%------------------------------------------------------------------------------
% File     : ITP276_1 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Uniqueness 00216_013627
% 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    : 0075_VEBT_Uniqueness_00216_013627 [Des22]

% Status   : ContradictoryAxioms
% Rating   : 0.44 v8.2.0, 0.20 v8.1.0
% Syntax   : Number of formulae    : 13312 (3824 unt;3041 typ;   0 def)
%            Number of atoms       : 26083 (10204 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 18633 (2821   ~; 516   |;2025   &)
%                                         (2421 <=>;10850  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   38 (   2 avg)
%            Number of types       :  640 ( 639 usr)
%            Number of type conns  : 2985 (1923   >;1062   *;   0   +;   0  <<)
%            Number of predicates  :  160 ( 157 usr;   2 prp; 0-4 aty)
%            Number of functors    : 2245 (2245 usr; 479 con; 0-8 aty)
%            Number of variables   : 26070 (25400   !; 670   ?;26070   :)
% 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 15:41:40.684
%------------------------------------------------------------------------------
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tff(ty_n_t__fun_It__Num__Onum_Mt__Num__Onum_J,type,
    fun_num_num: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Nat__Onat_J,type,
    fun_num_nat: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
    fun_num_int: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
    fun_nat_rat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Num__Onum_J,type,
    fun_nat_num: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    fun_nat_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    fun_nat_int: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
    fun_int_rat: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Num__Onum_J,type,
    fun_int_num: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
    fun_int_nat: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    fun_int_int: $tType ).

tff(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
    set_complex: $tType ).

tff(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
    filter_real: $tType ).

tff(ty_n_t__Option__Ooption_It__HOL__Obool_J,type,
    option_bool: $tType ).

tff(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
    option_num: $tType ).

tff(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
    option_nat: $tType ).

tff(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
    option_int: $tType ).

tff(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
    filter_nat: $tType ).

tff(ty_n_t__Set__Oset_It__String__Ochar_J,type,
    set_char: $tType ).

tff(ty_n_t__List__Olist_It__Real__Oreal_J,type,
    list_real: $tType ).

tff(ty_n_t__List__Olist_It__HOL__Obool_J,type,
    list_bool: $tType ).

tff(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
    set_real: $tType ).

tff(ty_n_t__List__Olist_It__Num__Onum_J,type,
    list_num: $tType ).

tff(ty_n_t__List__Olist_It__Nat__Onat_J,type,
    list_nat: $tType ).

tff(ty_n_t__List__Olist_It__Int__Oint_J,type,
    list_int: $tType ).

tff(ty_n_t__VEBT____Definitions__OVEBT,type,
    vEBT_VEBT: $tType ).

tff(ty_n_t__Set__Oset_It__HOL__Obool_J,type,
    set_bool: $tType ).

tff(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
    set_rat: $tType ).

tff(ty_n_t__Set__Oset_It__Num__Onum_J,type,
    set_num: $tType ).

tff(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

tff(ty_n_t__Set__Oset_It__Int__Oint_J,type,
    set_int: $tType ).

tff(ty_n_t__Code____Numeral__Onatural,type,
    code_natural: $tType ).

tff(ty_n_t__Code____Numeral__Ointeger,type,
    code_integer: $tType ).

tff(ty_n_t__Extended____Nat__Oenat,type,
    extended_enat: $tType ).

tff(ty_n_t__Typerep__Otyperep,type,
    typerep: $tType ).

tff(ty_n_t__Complex__Ocomplex,type,
    complex: $tType ).

tff(ty_n_t__String__Oliteral,type,
    literal: $tType ).

tff(ty_n_t__String__Ochar,type,
    char: $tType ).

tff(ty_n_t__Real__Oreal,type,
    real: $tType ).

tff(ty_n_t__HOL__Obool,type,
    bool: $tType ).

tff(ty_n_t__Rat__Orat,type,
    rat: $tType ).

tff(ty_n_t__Num__Onum,type,
    num: $tType ).

tff(ty_n_t__Nat__Onat,type,
    nat: $tType ).

tff(ty_n_t__Int__Oint,type,
    int: $tType ).

% Explicit typings (2402)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: set_complex > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__aaa____,type,
    aTP_Lamm_aaa: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__aab____,type,
    aTP_Lamm_aab: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__aac____,type,
    aTP_Lamm_aac: extended_enat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__aad____,type,
    aTP_Lamm_aad: nat > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_ATP_058Lamm__aae____,type,
    aTP_Lamm_aae: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__aaf____,type,
    aTP_Lamm_aaf: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__aag____,type,
    aTP_Lamm_aag: fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__aah____,type,
    aTP_Lamm_aah: nat > fun_na5042103099923536755t_bool ).

tff(sy_c_ATP_058Lamm__aai____,type,
    aTP_Lamm_aai: fun_fun_nat_rat_bool ).

tff(sy_c_ATP_058Lamm__aaj____,type,
    aTP_Lamm_aaj: fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__aak____,type,
    aTP_Lamm_aak: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__aal____,type,
    aTP_Lamm_aal: fun_nat_int ).

tff(sy_c_ATP_058Lamm__aam____,type,
    aTP_Lamm_aam: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aan____,type,
    aTP_Lamm_aan: code_natural > fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__aao____,type,
    aTP_Lamm_aao: fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__aap____,type,
    aTP_Lamm_aap: code_natural > fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_list_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: set_set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: set_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: vEBT_VEBT > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: vEBT_VEBT > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_real_real ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_int_int ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_real_real ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_int_int ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: fun_nat_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: set_nat > fun_nat_fun_nat_bool ).

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

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: set_bool > fun_na205931550456536010l_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_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: set_list_nat > fun_na5915592536692849161t_bool ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: set_set_nat > fun_na5144080005386289891t_bool ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: set_bool > fun_na205931550456536010l_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_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: nat > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: nat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: set_real > fun_fu4085784932660848663l_bool ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: set_real > fun_fu4455227587713718890l_bool ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: set_nat > fun_fu9135849066224463439t_bool ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: set_nat > fun_fu8715438531829836670t_bool ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: set_int > fun_fu4435397439461041559t_bool ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: set_int > fun_fu5963139849246984042t_bool ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: set_complex > fun_fu8557032673370904723x_bool ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: set_complex > fun_fu9203097815590270564x_bool ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: set_real > fun_fu914841811607101702l_bool ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: set_real > fun_fu7100649987335273288l_bool ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: set_nat > fun_fu4346542433904371390t_bool ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: set_nat > fun_fu8382797250367883996t_bool ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: set_int > fun_fu7152481525987436038t_bool ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: set_int > fun_fu5129045444369776328t_bool ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: set_complex > fun_fu5519326811792979714x_bool ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: set_complex > fun_fu3191020621175015362x_bool ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: set_real > fun_fu914841811607101702l_bool ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: set_real > fun_fu7100649987335273288l_bool ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: set_nat > fun_fu4346542433904371390t_bool ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: set_nat > fun_fu8382797250367883996t_bool ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: set_int > fun_fu7152481525987436038t_bool ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: set_int > fun_fu5129045444369776328t_bool ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: set_complex > fun_fu5519326811792979714x_bool ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: set_complex > fun_fu3191020621175015362x_bool ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: set_complex > fun_fu5995210064384465734x_bool ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: set_real > fun_fu2277326744902863452l_bool ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: set_real > fun_fu6750725090257165812l_bool ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: set_nat > fun_fu3052598110916186900t_bool ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: set_nat > fun_fu2971308621445215880t_bool ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_nat_real ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: int > fun_fu1008360527747082367nt_rat ).

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

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: rat > fun_complex_rat ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: rat > fun_int_rat ).

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

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: real > fun_int_real ).

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

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: int > fun_complex_int ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: nat > fun_complex_nat ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: nat > fun_int_nat ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: fun_nat_fun_nat_nat ).

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

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

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

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: int > fun_fu1143798476471108391nt_int ).

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

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_nat_nat > fun_fu2454420574033104442at_nat ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_int_int > fun_fu923337204662071120nt_int ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_int_int > fun_fu3739269683040691718nt_int ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: fun_nat_real > fun_fu3306995184869731826t_real ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: set_real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_real_complex > fun_fu1393723446858488622omplex ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: set_nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_nat_complex > fun_fu7905158015251895726omplex ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: set_int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_int_complex > fun_fu6363802315473983150omplex ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: set_complex > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: fun_complex_complex > fun_fu248857536038387758omplex ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_complex_real > fun_fu1911030855436031788x_real ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_real_rat > fun_fu7249360421009165912al_rat ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_nat_rat > fun_fu2739651132436891864at_rat ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_int_rat > fun_fu2671659548940339672nt_rat ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_real_complex > fun_se2317959235522472630omplex ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_int_complex > fun_se7807923622652057142omplex ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: fun_complex_complex > fun_se7512233253626865466omplex ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_real_real > fun_se5601468920065002932l_real ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: fun_int_real > fun_se7886327339988957492t_real ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_complex_real > fun_se7589301288977092664x_real ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_real_rat > fun_se2428050584324877792al_rat ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_int_rat > fun_se937733209267607904nt_rat ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_complex_rat > fun_se8588693959369716324ex_rat ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_real_nat > fun_se8808036921802555352al_nat ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_real_complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_int_complex > fun_int_bool ).

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

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_real_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_int_real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: fun_complex_real > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_real_rat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_int_rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_complex_rat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_real_nat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_nat_real > fun_na7718645245721260691t_real ).

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_nat_int > fun_nat_int ).

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_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: code_integer > fun_Co6226381647958292740nteger ).

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

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: extended_enat > fun_Ex2441982661509597004d_enat ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: complex > fun_co3258106519541737638omplex ).

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

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

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: real > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: fun_nat_real ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_nat_int ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: nat > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: set_nat > fun_fu40286998686754419omplex ).

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

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_nat_bool > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_nat_bool > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_nat_bool > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_nat_bool > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_complex_nat > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_nat_nat > fun_fu7002938529518989751at_nat ).

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

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

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

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

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

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: set_real > fun_fu6442698403317780995l_bool ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: set_real > fun_fu6442698403317780995l_bool ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: set_real > fun_fu8273873231336023293al_nat ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: set_real > fun_fu4470565429337120259l_bool ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: set_nat > fun_fu9073671928554219267t_bool ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: set_nat > fun_fu6241273384353077465al_nat ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: set_real > fun_fu66453571216240387l_bool ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: set_int > fun_fu3139726878529933315t_bool ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: set_int > fun_fu5144190403709078909al_nat ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: set_real > fun_fu6440467031773438595l_bool ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: set_complex > fun_fu4189668387880940419x_bool ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: set_complex > fun_fu7828569628974167419al_nat ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: set_int > fun_fu7145108737927999747t_bool ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: set_real > fun_fu4071835430614306819l_bool ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: set_real > fun_fu4437885374841300733nt_nat ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: set_int > fun_fu7719728765329718531t_bool ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: set_nat > fun_fu2583778531842586371t_bool ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: set_nat > fun_fu2415648189221275865nt_nat ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: set_int > fun_fu691032689367502339t_bool ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: set_int > fun_fu691032689367502339t_bool ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: set_int > fun_fu5453490273685409661nt_nat ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: set_int > fun_fu8949065344085950851t_bool ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: set_complex > fun_fu3391973010043818627x_bool ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: set_complex > fun_fu1675835286861305467nt_nat ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: set_complex > fun_fu420965002030662787x_bool ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: set_real > fun_fu2671763645923160963l_bool ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: set_real > fun_fu2111190393460728701ex_nat ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: set_complex > fun_fu1598310740840436355x_bool ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: set_nat > fun_fu5088574307552523139t_bool ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: set_nat > fun_fu2009456248099990617ex_nat ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: real > fun_fu6899955116716589935t_real ).

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

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: complex > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: int > fun_nat_fun_int_int ).

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

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_real_real ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_int_int ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_nat_Code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: fun_nat_real > fun_na6724076748692969250l_real ).

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

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: fun_nat_int > fun_na7654559027617327379at_int ).

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

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

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

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: complex > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: int > fun_fu4328765270694171897t_real ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: real > fun_fu9132278742410140537l_real ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: complex > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: int > fun_fu1617438229336688381nt_rat ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: real > fun_fu7187499828406537085al_rat ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: complex > fun_fu2513798379308225899ex_nat ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: int > fun_fu973771059875708133nt_nat ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: real > fun_fu4705247872303406806omplex ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: nat > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: int > fun_fu9139781051346917846omplex ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: complex > fun_fu8998703528969179354omplex ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: real > fun_fu8560815523039441616l_real ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: nat > fun_fu8064956510386623256t_real ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: int > fun_fu4705516758182030288t_real ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: complex > fun_fu5754211913296672468x_real ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: real > fun_fu2940912951466865492al_rat ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: nat > fun_fu4710970720072501148at_rat ).

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

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: real > fun_na5225841814499866095t_real ).

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

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

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

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

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: int > fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: code_integer > fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: rat > fun_nat_fun_nat_rat ).

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

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

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_nat_real > fun_nat_real ).

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

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

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

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: set_list_nat > fun_na5915592536692849161t_bool ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: set_set_nat > fun_na5144080005386289891t_bool ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_complex_real > fun_fu1911030855436031788x_real ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_real_rat > fun_fu7249360421009165912al_rat ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: fun_int_rat > fun_fu2671659548940339672nt_rat ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_complex_rat > fun_fu4947971316162376536ex_rat ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_nat_rat > fun_fu2739651132436891864at_rat ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_real_nat > fun_fu4405974721632067664al_nat ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: fun_int_nat > fun_fu9051645886418017232nt_nat ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_complex_nat > fun_fu2104585616785278288ex_nat ).

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

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: fun_int_bool > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_complex_bool > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_real_bool > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_int_bool > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_complex_bool > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_nat_bool > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_real_bool > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: fun_int_bool > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_complex_bool > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_nat_real > fun_re4238438383274954274t_real ).

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

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

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

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

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

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

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: fun_nat_fun_nat_int > fun_nat_int ).

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

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

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

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

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_nat_complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: fun_nat_real > fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: fun_nat_rat > fun_fu6228432884689642900at_rat ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: fun_nat_rat > fun_fu4710970720072501148at_rat ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_nat_real > fun_fu8064956510386623256t_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: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_nat_nat > fun_fu4413992040155564420at_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_fun_nat_nat ).

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

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

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: nat > fun_fu960360787657196261at_int ).

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

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

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: nat > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: nat > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: nat > fun_fu5935951186182374353at_nat ).

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

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

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_nat_complex > fun_co3194742168270691230omplex ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: fun_nat_rat > fun_ra2740371568465858076at_rat ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: fun_nat_int > fun_in281396392327853336at_int ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: fun_nat_real > fun_re7819166435457029784t_real ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: set_list_nat > fun_na5915592536692849161t_bool ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: set_set_nat > fun_na5144080005386289891t_bool ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: fun_nat_complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: fun_nat_rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_nat_int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: fun_nat_real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: nat > fun_se7483391254894679335x_bool ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: nat > fun_se3848302202764083907t_bool ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: nat > fun_se1818217329951269751t_bool ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: nat > fun_se5463523874219726535T_bool ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: nat > fun_se2024411615333173255l_bool ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: nat > fun_se7165406737953478955t_bool ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: nat > fun_se5958083809616857059t_bool ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: nat > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_real_bool > fun_fu9132278742410140537l_real ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: fun_int_bool > fun_fu4328765270694171897t_real ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_complex_bool > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_nat_bool > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_real_bool > fun_fu7187499828406537085al_rat ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_int_bool > fun_fu1617438229336688381nt_rat ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: fun_complex_bool > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_nat_bool > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: fun_real_bool > fun_fu3768230810189712741al_nat ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: fun_int_bool > fun_fu973771059875708133nt_nat ).

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

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: complex > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: nat > fun_in1521351555827773387at_int ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: nat > fun_co6147893885205795665omplex ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: nat > fun_Co8502721031311864414nteger ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: nat > fun_ra3995598502166916815at_rat ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: nat > fun_re8542384036533109323t_real ).

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

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

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

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: fun_nat_fun_nat_bool ).

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

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: real > fun_fu3477889739401271834t_real ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: real > fun_fu4806068833880117316t_real ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: real > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: real > fun_fu4540635375566031512t_real ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_nat_real ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: fun_int_int > fun_int_int ).

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

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_co8153928105989009288nteger > fun_fu1400560744564786701nteger ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: fun_complex_rat > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: fun_nat_real ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: fun_real_bool ).

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

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: fun_na819858167508000762st_nat ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: rat > fun_in1696471567231710611nt_int ).

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_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: fun_real_real ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: fun_real_real > fun_re7681423839302404990l_real ).

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

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__ym____,type,
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tff(sy_c_ATP_058Lamm__yn____,type,
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tff(sy_c_ATP_058Lamm__yo____,type,
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tff(sy_c_ATP_058Lamm__yp____,type,
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tff(sy_c_ATP_058Lamm__yq____,type,
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tff(sy_c_ATP_058Lamm__yr____,type,
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tff(sy_c_ATP_058Lamm__ys____,type,
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tff(sy_c_ATP_058Lamm__yt____,type,
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tff(sy_c_ATP_058Lamm__yu____,type,
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tff(sy_c_ATP_058Lamm__yv____,type,
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tff(sy_c_ATP_058Lamm__yw____,type,
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tff(sy_c_ATP_058Lamm__yx____,type,
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tff(sy_c_ATP_058Lamm__yy____,type,
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tff(sy_c_ATP_058Lamm__yz____,type,
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tff(sy_c_ATP_058Lamm__za____,type,
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tff(sy_c_ATP_058Lamm__zb____,type,
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tff(sy_c_ATP_058Lamm__zc____,type,
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tff(sy_c_ATP_058Lamm__zd____,type,
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tff(sy_c_ATP_058Lamm__ze____,type,
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tff(sy_c_ATP_058Lamm__zf____,type,
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tff(sy_c_ATP_058Lamm__zg____,type,
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tff(sy_c_ATP_058Lamm__zh____,type,
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tff(sy_c_ATP_058Lamm__zi____,type,
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tff(sy_c_ATP_058Lamm__zj____,type,
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tff(sy_c_ATP_058Lamm__zk____,type,
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tff(sy_c_ATP_058Lamm__zl____,type,
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tff(sy_c_ATP_058Lamm__zm____,type,
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tff(sy_c_ATP_058Lamm__zn____,type,
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tff(sy_c_ATP_058Lamm__zo____,type,
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tff(sy_c_ATP_058Lamm__zp____,type,
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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_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
    lattic8265883725875713057ax_nat: fun_set_nat_nat ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    lattic5364784637807008409ex_nat: ( fun_complex_nat * set_complex ) > complex ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Num__Onum,type,
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tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Rat__Orat,type,
    lattic4729654577720512673ex_rat: ( fun_complex_rat * set_complex ) > complex ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Num__Onum,type,
    lattic5003618458639192673nt_num: ( fun_int_num * set_int ) > int ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Rat__Orat,type,
    lattic7811156612396918303nt_rat: ( fun_int_rat * set_int ) > int ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
    lattic7446932960582359483at_nat: fun_nat_nat > fun_set_nat_nat ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Num__Onum,type,
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tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Rat__Orat,type,
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tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Num__Onum,type,
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tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Rat__Orat,type,
    lattic4420706379359479199al_rat: ( fun_real_rat * set_real ) > real ).

tff(sy_c_Lattices__Big_Osemilattice__neutr__set_OF_001t__Nat__Onat,type,
    lattic7826324295020591184_F_nat: ( fun_nat_fun_nat_nat * nat ) > fun_set_nat_nat ).

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

tff(sy_c_List_Oappend_001t__Nat__Onat,type,
    append_nat: ( list_nat * list_nat ) > list_nat ).

tff(sy_c_List_Oconcat_001t__HOL__Obool,type,
    concat_bool: list_list_bool > list_bool ).

tff(sy_c_List_Oconcat_001t__Int__Oint,type,
    concat_int: list_list_int > list_int ).

tff(sy_c_List_Oconcat_001t__Nat__Onat,type,
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tff(sy_c_List_Oconcat_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Ocount__list_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Ocount__list_001t__HOL__Obool,type,
    count_list_bool: list_bool > fun_bool_nat ).

tff(sy_c_List_Ocount__list_001t__Int__Oint,type,
    count_list_int: list_int > fun_int_nat ).

tff(sy_c_List_Ocount__list_001t__Nat__Onat,type,
    count_list_nat: list_nat > fun_nat_nat ).

tff(sy_c_List_Ocount__list_001t__Real__Oreal,type,
    count_list_real: list_real > fun_real_nat ).

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_Odistinct_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Odistinct_001t__HOL__Obool,type,
    distinct_bool: fun_list_bool_bool ).

tff(sy_c_List_Odistinct_001t__Int__Oint,type,
    distinct_int: fun_list_int_bool ).

tff(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
    distinct_list_nat: list_list_nat > $o ).

tff(sy_c_List_Odistinct_001t__Nat__Onat,type,
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tff(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Odistinct_001t__Real__Oreal,type,
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tff(sy_c_List_Odistinct_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Odistinct_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oenumerate_001t__HOL__Obool,type,
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tff(sy_c_List_Oenumerate_001t__Int__Oint,type,
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tff(sy_c_List_Oenumerate_001t__Nat__Onat,type,
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tff(sy_c_List_Oenumerate_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Ofind_001t__HOL__Obool,type,
    find_bool: ( fun_bool_bool * list_bool ) > option_bool ).

tff(sy_c_List_Ofind_001t__Int__Oint,type,
    find_int: ( fun_int_bool * list_int ) > option_int ).

tff(sy_c_List_Ofind_001t__Nat__Onat,type,
    find_nat: ( fun_nat_bool * list_nat ) > option_nat ).

tff(sy_c_List_Ofind_001t__Num__Onum,type,
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tff(sy_c_List_Ofind_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
    fold_nat_nat: ( fun_nat_fun_nat_nat * list_nat ) > fun_nat_nat ).

tff(sy_c_List_Olenlex_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_List_Olenlex_001t__Int__Oint,type,
    lenlex_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).

tff(sy_c_List_Olenlex_001t__Nat__Onat,type,
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tff(sy_c_List_Olex_001t__HOL__Obool,type,
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tff(sy_c_List_Olex_001t__Int__Oint,type,
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tff(sy_c_List_Olex_001t__Nat__Onat,type,
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tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
    linord2614967742042102400et_nat: set_nat > list_nat ).

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tff(sy_c_List_Olist_OCons_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Olist_OCons_001t__HOL__Obool,type,
    cons_bool: ( bool * list_bool ) > list_bool ).

tff(sy_c_List_Olist_OCons_001t__Int__Oint,type,
    cons_int: ( int * list_int ) > list_int ).

tff(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
    cons_nat: ( nat * list_nat ) > list_nat ).

tff(sy_c_List_Olist_OCons_001t__Num__Onum,type,
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tff(sy_c_List_Olist_OCons_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_OCons_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Olist_OCons_001t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Olist_Oset_001t__HOL__Obool,type,
    set_bool2: list_bool > set_bool ).

tff(sy_c_List_Olist_Oset_001t__Int__Oint,type,
    set_int2: list_int > set_int ).

tff(sy_c_List_Olist_Oset_001t__List__Olist_It__HOL__Obool_J,type,
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tff(sy_c_List_Olist_Oset_001t__List__Olist_It__Int__Oint_J,type,
    set_list_int2: list_list_int > set_list_int ).

tff(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
    set_list_nat2: list_list_nat > set_list_nat ).

tff(sy_c_List_Olist_Oset_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
    set_nat2: list_nat > set_nat ).

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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    set_Pr5999470521830281550l_real: list_P8689742595348180415l_real > set_Pr6218003697084177305l_real ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
    set_Pr8897343066327330088T_VEBT: list_P877281246627933069T_VEBT > set_Pr6019664923565264691T_VEBT ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    set_Pr9040384385603167362et_nat: list_P6254988961118846195et_nat > set_Pr5488025237498180813et_nat ).

tff(sy_c_List_Olist_Oset_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,
    set_Pr3765526544606949372at_nat: list_P5464809261938338413at_nat > set_Pr4329608150637261639at_nat ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Complex__Ocomplex_J,type,
    set_Pr6387300694196750780omplex: list_P4108580160459392801omplex > set_Pr216944050393469383omplex ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J,type,
    set_Pr3399341499479959199T_bool: list_P44034876932920720T_bool > set_Pr2229214029182756970T_bool ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    set_Pr2853735649769556538BT_int: list_P4547456442757143711BT_int > set_Pr5066593544530342725BT_int ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    set_Pr7031586669278753246BT_nat: list_P7037539587688870467BT_nat > set_Pr7556676689462069481BT_nat ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    set_Pr1087130671499945274T_real: list_P2623026923184700063T_real > set_Pr7765410600122031685T_real ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    set_Pr9182192707038809660T_VEBT: list_P7413028617227757229T_VEBT > set_Pr6192946355708809607T_VEBT ).

tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
    set_real2: list_real > set_real ).

tff(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
    set_set_nat2: list_set_nat > set_set_nat ).

tff(sy_c_List_Olist_Oset_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_se5049602875457034614at_nat: list_s1210847774152347623at_nat > set_se7855581050983116737at_nat ).

tff(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
    set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).

tff(sy_c_List_Olist_Osize__list_001t__Typerep__Otyperep,type,
    size_list_typerep: ( fun_typerep_nat * list_typerep ) > nat ).

tff(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
    size_list_VEBT_VEBT: ( fun_VEBT_VEBT_nat * list_VEBT_VEBT ) > nat ).

tff(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_nat ).

tff(sy_c_List_Olist__update_001t__Code____Numeral__Ointeger,type,
    list_u5447711078246177391nteger: ( list_Code_integer * nat * code_integer ) > list_Code_integer ).

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

tff(sy_c_List_Olist__update_001t__HOL__Obool,type,
    list_update_bool: ( list_bool * nat * bool ) > list_bool ).

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

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

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    list_u2254550707601501961nteger: ( list_P5578671422887162913nteger * nat * produc8923325533196201883nteger ) > list_P5578671422887162913nteger ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    list_u3002344382305578791nt_int: ( list_P5707943133018811711nt_int * nat * product_prod_int_int ) > list_P5707943133018811711nt_int ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    list_u6180841689913720943at_nat: ( list_P6011104703257516679at_nat * nat * product_prod_nat_nat ) > list_P6011104703257516679at_nat ).

tff(sy_c_List_Olist__update_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,
    list_u5003261594476800725at_nat: ( list_P8469869581646625389at_nat * nat * produc859450856879609959at_nat ) > list_P8469869581646625389at_nat ).

tff(sy_c_List_Olist__update_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,
    list_u4696772448584712917at_nat: ( list_P5464809261938338413at_nat * nat * produc3843707927480180839at_nat ) > list_P5464809261938338413at_nat ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    list_u6961636818849549845T_VEBT: ( list_P7413028617227757229T_VEBT * nat * produc8243902056947475879T_VEBT ) > list_P7413028617227757229T_VEBT ).

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

tff(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
    list_update_set_nat: ( list_set_nat * nat * set_nat ) > list_set_nat ).

tff(sy_c_List_Olist__update_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    list_u8444657558853818831at_nat: ( list_s1210847774152347623at_nat * nat * set_Pr1261947904930325089at_nat ) > list_s1210847774152347623at_nat ).

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

tff(sy_c_List_Olistrel_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    listre5734910445319291053nteger: set_Pr4811707699266497531nteger > set_Pr7565137564259432987nteger ).

tff(sy_c_List_Olistrel_001t__HOL__Obool_001t__HOL__Obool,type,
    listrel_bool_bool: set_Pr8621012493645324487l_bool > set_Pr4016434106243378759t_bool ).

tff(sy_c_List_Olistrel_001t__HOL__Obool_001t__Int__Oint,type,
    listrel_bool_int: set_Pr4651914117858333608ol_int > set_Pr7659629972148914366st_int ).

tff(sy_c_List_Olistrel_001t__HOL__Obool_001t__Nat__Onat,type,
    listrel_bool_nat: set_Pr7141997262790060364ol_nat > set_Pr3249806354082057314st_nat ).

tff(sy_c_List_Olistrel_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    listre4049385574073087438T_VEBT: set_Pr8097748915547837924T_VEBT > set_Pr3294270863102529764T_VEBT ).

tff(sy_c_List_Olistrel_001t__Int__Oint_001t__Int__Oint,type,
    listrel_int_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).

tff(sy_c_List_Olistrel_001t__Nat__Onat_001t__HOL__Obool,type,
    listrel_nat_bool: set_Pr4681767519595310322t_bool > set_Pr1340933433341948520t_bool ).

tff(sy_c_List_Olistrel_001t__Nat__Onat_001t__Nat__Onat,type,
    listrel_nat_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

tff(sy_c_List_Olistrel_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    listre5761932458788874033T_VEBT: set_Pr6167073792073659919T_VEBT > set_Pr1262583345697558789T_VEBT ).

tff(sy_c_List_Olistrel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    listre818007680106770737at_nat: set_Pr8693737435421807431at_nat > set_Pr1542805901266377927at_nat ).

tff(sy_c_List_Olistrel_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    listre2047417242196832561at_nat: set_Pr4329608150637261639at_nat > set_Pr4333006031979791559at_nat ).

tff(sy_c_List_Olistrel_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
    listre7233207804915621076T_bool: set_Pr2229214029182756970T_bool > set_Pr5897181536329667306t_bool ).

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

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

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

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

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

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

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

tff(sy_c_List_Omeasures_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    measur2694323259624372065at_nat: list_f2430256441796990922at_nat > set_Pr4329608150637261639at_nat ).

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

tff(sy_c_List_Onth_001t__Complex__Ocomplex,type,
    nth_complex: list_complex > fun_nat_complex ).

tff(sy_c_List_Onth_001t__HOL__Obool,type,
    nth_bool: list_bool > fun_nat_bool ).

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

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

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

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__HOL__Obool_Mt__HOL__Obool_J,type,
    nth_Pr2856333732135533678l_bool: ( list_P2722043638750504429l_bool * nat ) > produc2804947525362035943l_bool ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__HOL__Obool_Mt__Int__Oint_J,type,
    nth_Pr7231943226834047147ol_int: ( list_P6751438739536979202ol_int * nat ) > produc5612625445546783218ol_int ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__HOL__Obool_Mt__Nat__Onat_J,type,
    nth_Pr2186422209488468047ol_nat: ( list_P18149847613930150ol_nat * nat ) > produc567104428201204118ol_nat ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__HOL__Obool_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__HOL__Obool_J,type,
    nth_Pr396318242284072693t_bool: ( list_P6781292141273955916t_bool * nat ) > produc8000372497851584572t_bool ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr744662078594809490T_VEBT: ( list_P5647936690300460905T_VEBT * nat ) > produc8025551001238799321T_VEBT ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J,type,
    nth_Pr1289831921885633937T_bool: ( list_P44034876932920720T_bool * nat ) > produc6448083994118892042T_bool ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    nth_Pr6837108013167703752BT_int: ( list_P4547456442757143711BT_int * nat ) > produc4894624898956917775BT_int ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    nth_Pr1791586995822124652BT_nat: ( list_P7037539587688870467BT_nat * nat ) > produc9072475918466114483BT_nat ).

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

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

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

tff(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__HOL__Obool,type,
    product_bool_bool: ( list_bool * list_bool ) > list_P2722043638750504429l_bool ).

tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__Int__Oint,type,
    product_bool_int: ( list_bool * list_int ) > list_P6751438739536979202ol_int ).

tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__Nat__Onat,type,
    product_bool_nat: ( list_bool * list_nat ) > list_P18149847613930150ol_nat ).

tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    produc2282853104913622818T_VEBT: ( list_bool * list_VEBT_VEBT ) > list_P5912569763298001674T_VEBT ).

tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__HOL__Obool,type,
    product_nat_bool: ( list_nat * list_bool ) > list_P6781292141273955916t_bool ).

tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc7156399406898700509T_VEBT: ( list_nat * list_VEBT_VEBT ) > list_P5647936690300460905T_VEBT ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
    produc5466675335756156456T_bool: ( list_VEBT_VEBT * list_bool ) > list_P44034876932920720T_bool ).

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

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

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

tff(sy_c_List_Oproduct__lists_001t__HOL__Obool,type,
    product_lists_bool: list_list_bool > list_list_bool ).

tff(sy_c_List_Oproduct__lists_001t__Int__Oint,type,
    product_lists_int: list_list_int > list_list_int ).

tff(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
    product_lists_nat: list_list_nat > list_list_nat ).

tff(sy_c_List_Oproduct__lists_001t__VEBT____Definitions__OVEBT,type,
    produc3021084454716106787T_VEBT: list_list_VEBT_VEBT > list_list_VEBT_VEBT ).

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

tff(sy_c_List_Oreplicate_001t__HOL__Obool,type,
    replicate_bool: ( nat * bool ) > list_bool ).

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

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

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

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

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

tff(sy_c_List_Orotate1_001t__HOL__Obool,type,
    rotate1_bool: list_bool > list_bool ).

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

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

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

tff(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
    sorted_wrt_nat: fun_nat_fun_nat_bool > fun_list_nat_bool ).

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

tff(sy_c_List_Otake_001t__VEBT____Definitions__OVEBT,type,
    take_VEBT_VEBT: ( nat * list_VEBT_VEBT ) > list_VEBT_VEBT ).

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

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

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

tff(sy_c_List_Ozip_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    zip_complex_complex: ( list_complex * list_complex ) > list_P7664491975274850627omplex ).

tff(sy_c_List_Ozip_001t__Complex__Ocomplex_001t__HOL__Obool,type,
    zip_complex_bool: ( list_complex * list_bool ) > list_P7115283683551655918x_bool ).

tff(sy_c_List_Ozip_001t__Complex__Ocomplex_001t__Int__Oint,type,
    zip_complex_int: ( list_complex * list_int ) > list_P2206113689347244737ex_int ).

tff(sy_c_List_Ozip_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    zip_complex_nat: ( list_complex * list_nat ) > list_P4696196834278971493ex_nat ).

tff(sy_c_List_Ozip_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    zip_complex_real: ( list_complex * list_real ) > list_P7647014805210017729x_real ).

tff(sy_c_List_Ozip_001t__Complex__Ocomplex_001t__VEBT____Definitions__OVEBT,type,
    zip_co9157518722488180109T_VEBT: ( list_complex * list_VEBT_VEBT ) > list_P7977503562704621835T_VEBT ).

tff(sy_c_List_Ozip_001t__HOL__Obool_001t__Complex__Ocomplex,type,
    zip_bool_complex: ( list_bool * list_complex ) > list_P7820136468425902468omplex ).

tff(sy_c_List_Ozip_001t__HOL__Obool_001t__HOL__Obool,type,
    zip_bool_bool: ( list_bool * list_bool ) > list_P2722043638750504429l_bool ).

tff(sy_c_List_Ozip_001t__HOL__Obool_001t__Int__Oint,type,
    zip_bool_int: ( list_bool * list_int ) > list_P6751438739536979202ol_int ).

tff(sy_c_List_Ozip_001t__HOL__Obool_001t__Nat__Onat,type,
    zip_bool_nat: ( list_bool * list_nat ) > list_P18149847613930150ol_nat ).

tff(sy_c_List_Ozip_001t__HOL__Obool_001t__Real__Oreal,type,
    zip_bool_real: ( list_bool * list_real ) > list_P8975979458798387586l_real ).

tff(sy_c_List_Ozip_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    zip_bool_VEBT_VEBT: ( list_bool * list_VEBT_VEBT ) > list_P5912569763298001674T_VEBT ).

tff(sy_c_List_Ozip_001t__Int__Oint_001t__Complex__Ocomplex,type,
    zip_int_complex: ( list_int * list_complex ) > list_P1797514011394873281omplex ).

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

tff(sy_c_List_Ozip_001t__Int__Oint_001t__Real__Oreal,type,
    zip_int_real: ( list_int * list_real ) > list_P6863124054624500543t_real ).

tff(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
    zip_nat_nat: ( list_nat * list_nat ) > list_P6011104703257516679at_nat ).

tff(sy_c_List_Ozip_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    zip_nat_VEBT_VEBT: ( list_nat * list_VEBT_VEBT ) > list_P5647936690300460905T_VEBT ).

tff(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    zip_Pr4664179122662387191at_nat: ( list_P6011104703257516679at_nat * list_P6011104703257516679at_nat ) > list_P8469869581646625389at_nat ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__Complex__Ocomplex,type,
    zip_real_complex: ( list_real * list_complex ) > list_P3881527313128557121omplex ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__HOL__Obool,type,
    zip_real_bool: ( list_real * list_bool ) > list_P3020538015149218416l_bool ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__Int__Oint,type,
    zip_real_int: ( list_real * list_int ) > list_P4344331454722006975al_int ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__Nat__Onat,type,
    zip_real_nat: ( list_real * list_nat ) > list_P6834414599653733731al_nat ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__Real__Oreal,type,
    zip_real_real: ( list_real * list_real ) > list_P8689742595348180415l_real ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    zip_real_VEBT_VEBT: ( list_real * list_VEBT_VEBT ) > list_P877281246627933069T_VEBT ).

tff(sy_c_List_Ozip_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    zip_set_nat_set_nat: ( list_set_nat * list_set_nat ) > list_P6254988961118846195et_nat ).

tff(sy_c_List_Ozip_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    zip_se5600341670672612855at_nat: ( list_s1210847774152347623at_nat * list_s1210847774152347623at_nat ) > list_P5464809261938338413at_nat ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Complex__Ocomplex,type,
    zip_VE2794733401258833515omplex: ( list_VEBT_VEBT * list_complex ) > list_P4108580160459392801omplex ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
    zip_VEBT_VEBT_bool: ( list_VEBT_VEBT * list_bool ) > list_P44034876932920720T_bool ).

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

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

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    zip_VEBT_VEBT_real: ( list_VEBT_VEBT * list_real ) > list_P2623026923184700063T_real ).

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

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

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

tff(sy_c_Nat_Onat_Ocase__nat_001t__HOL__Obool,type,
    case_nat_bool: ( bool * fun_nat_bool ) > fun_nat_bool ).

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

tff(sy_c_Nat_Onat_Opred,type,
    pred: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: fun_nat_Code_integer ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Onatural,type,
    semiri3763490453095760265atural: nat > code_natural ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: fun_nat_complex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
    semiri4216267220026989637d_enat: fun_na8797796813480665199d_enat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Code____Numeral__Ointeger,type,
    semiri4055485073559036834nteger: ( fun_Co8499238673658658103nteger * nat * code_integer ) > code_integer ).

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

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

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

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

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

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__HOL__Obool_J,type,
    size_size_list_bool: list_bool > nat ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__HOL__Obool_J_J,type,
    size_s9193230429364170767t_bool: list_list_bool > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
    size_s533118279054570080st_int: list_list_int > nat ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    size_s8217280938318005548T_VEBT: list_list_VEBT_VEBT > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: fun_list_nat_nat ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__HOL__Obool_Mt__HOL__Obool_J_J,type,
    size_s6988337448923680857l_bool: list_P2722043638750504429l_bool > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__HOL__Obool_Mt__Int__Oint_J_J,type,
    size_s753689718144262038ol_int: list_P6751438739536979202ol_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__HOL__Obool_Mt__Nat__Onat_J_J,type,
    size_s3243772863075988794ol_nat: list_P18149847613930150ol_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__HOL__Obool_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s7211631430880329078T_VEBT: list_P5912569763298001674T_VEBT > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__HOL__Obool_J_J,type,
    size_s783543119881238752t_bool: list_P6781292141273955916t_bool > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J_J,type,
    size_s1343096544515248124T_bool: list_P44034876932920720T_bool > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
    size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    size_s8736152011456118867at_nat: list_s1210847774152347623at_nat > nat ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
    size_size_num: fun_num_nat ).

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__Typerep__Otyperep,type,
    size_size_typerep: fun_typerep_nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
    size_size_VEBT_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_Nat__Bijection_Oint__decode,type,
    nat_int_decode: fun_nat_int ).

tff(sy_c_Nat__Bijection_Oint__encode,type,
    nat_int_encode: fun_int_nat ).

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_Osum__decode,type,
    nat_sum_decode: fun_na3635074868517587788at_nat ).

tff(sy_c_Nat__Bijection_Osum__encode,type,
    nat_sum_encode: fun_Su884838341992399436at_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_Onat__of__num,type,
    nat_of_num: fun_num_nat ).

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__Code____Numeral__Onatural,type,
    numera5444537566228673987atural: num > code_natural ).

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: fun_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__HOL__Obool,type,
    some_bool: bool > option_bool ).

tff(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
    some_int: int > option_int ).

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_OSome_001t__VEBT____Definitions__OVEBT,type,
    some_VEBT_VEBT: vEBT_VEBT > option_VEBT_VEBT ).

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 ) > fun_op2627489687811228755t_bool ).

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_Option_Ooption_Othe_001t__Num__Onum,type,
    the_num: option_num > num ).

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

tff(sy_c_Order__Relation_Olinear__order__on_001t__Nat__Onat,type,
    order_4473980167227706203on_nat: ( set_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_Order__Relation_Opartial__order__on_001t__Nat__Onat,type,
    order_5251275573222108571on_nat: ( set_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_Order__Relation_Opreorder__on_001t__Nat__Onat,type,
    order_4861654808422542329on_nat: ( set_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
    order_underS_nat: set_Pr1261947904930325089at_nat > fun_nat_set_nat ).

tff(sy_c_Order__Relation_Owell__order__on_001t__Nat__Onat,type,
    order_2888998067076097458on_nat: ( set_nat * set_Pr1261947904930325089at_nat ) > $o ).

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__HOL__Obool_J,type,
    bot_bot_set_bool: set_bool ).

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__List__Olist_It__Nat__Onat_J_J,type,
    bot_bot_set_list_nat: set_list_nat ).

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__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).

tff(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
    ord_Least_nat: fun_nat_bool > nat ).

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 * extended_enat ) > $o ).

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__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__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 > fun_set_real_bool ).

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
    ord_le3102999989581377725nteger: fun_Co8310303586393106575r_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Onatural,type,
    ord_le1926595141338095240atural: ( code_natural * code_natural ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
    ord_le2932123472753598470d_enat: ( extended_enat * extended_enat ) > $o ).

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_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__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__List__Olist_It__Nat__Onat_J_J,type,
    ord_le6045566169113846134st_nat: ( set_list_nat * set_list_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > fun_set_nat_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 * set_Pr1261947904930325089at_nat ) > $o ).

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_eq_set_real: set_real > fun_set_real_bool ).

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,
    ord_ma741700101516333627d_enat: ( extended_enat * extended_enat ) > extended_enat ).

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: fun_num_fun_num_num ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
    ord_max_rat: fun_rat_fun_rat_rat ).

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

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_max_set_nat: fun_se1909708154429037836et_nat ).

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

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
    order_Greatest_nat: fun_nat_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_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_5726023648592871131at_nat: fun_nat_nat > $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,
    top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Onatural,type,
    power_7079662738309270450atural: ( code_natural * nat ) > code_natural ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Extended____Nat__Oenat,type,
    power_8040749407984259932d_enat: ( extended_enat * nat ) > extended_enat ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

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

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Onatural_001t__Code____Numeral__Onatural,type,
    produc3574140220909816553atural: ( code_natural * code_natural ) > produc7822875418678951345atural ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    produc6639722614265839536atural: code_natural > fun_Pr7769407664385210434atural ).

tff(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    produc101793102246108661omplex: ( complex * complex ) > produc4411394909380815293omplex ).

tff(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__HOL__Obool,type,
    produc7867974272183215760x_bool: ( complex * bool ) > produc7783722241732527326x_bool ).

tff(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__Int__Oint,type,
    produc1367138851071493491ex_int: ( complex * int ) > produc6845221339535797307ex_int ).

tff(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    produc1369629321580543767ex_nat: ( complex * nat ) > produc1799700322190218207ex_nat ).

tff(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    produc1746590499379883635x_real: ( complex * real ) > produc8892588492097263291x_real ).

tff(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__VEBT____Definitions__OVEBT,type,
    produc2757191886755552429T_VEBT: ( complex * vEBT_VEBT ) > produc7151440242714718331T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Complex__Ocomplex,type,
    produc5418556796337449326omplex: ( bool * complex ) > produc8846855491279061364omplex ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__HOL__Obool,type,
    produc8297744024737662807l_bool: ( bool * bool ) > produc2804947525362035943l_bool ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Int__Oint,type,
    produc8871513391732212460ol_int: ( bool * int ) > produc5612625445546783218ol_int ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Nat__Onat,type,
    produc8874003862241262736ol_nat: ( bool * nat ) > produc567104428201204118ol_nat ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Real__Oreal,type,
    produc7587643690356850540l_real: ( bool * real ) > produc5853794331844391154l_real ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    produc9074556814250467700T_VEBT: ( bool * vEBT_VEBT ) > produc8408751633324799108T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Complex__Ocomplex,type,
    produc7948753499206759283omplex: ( int * complex ) > produc5838698208256999739omplex ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: ( int * int ) > product_prod_int_int ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Real__Oreal,type,
    produc801115645435158769t_real: ( int * real ) > produc679980390762269497t_real ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__Code____Numeral__Ointeger_J_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    produc750622340256944499nteger: ( list_Code_integer * list_Code_integer ) > produc862207588354017979nteger ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__HOL__Obool_J_001t__List__Olist_It__HOL__Obool_J,type,
    produc970142415094439895t_bool: ( list_bool * list_bool ) > produc7619963554364372839t_bool ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__HOL__Obool_J_001t__List__Olist_It__Int__Oint_J,type,
    produc7189171366127010562st_int: ( list_bool * list_int ) > produc1591970419219485960st_int ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__HOL__Obool_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc2143650348781431462st_nat: ( list_bool * list_nat ) > produc4082053564151212716st_nat ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__HOL__Obool_J_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    produc6468702466802804596T_VEBT: ( list_bool * list_VEBT_VEBT ) > produc8209973772417936516T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
    produc364263696895485585st_int: ( list_int * list_int ) > produc1186641810826059865st_int ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__HOL__Obool_J,type,
    produc5531179457150407524t_bool: ( list_nat * list_bool ) > produc203981762660283314t_bool ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc2694037385005941721st_nat: ( list_nat * list_nat ) > produc1828647624359046049st_nat ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    produc8335345208264861441T_VEBT: ( list_nat * list_VEBT_VEBT ) > produc872621073311890639T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc5943733680697469783at_nat: ( list_P6011104703257516679at_nat * list_P6011104703257516679at_nat ) > produc6392793444374437607at_nat ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc7536900900485677911at_nat: ( list_s1210847774152347623at_nat * list_s1210847774152347623at_nat ) > produc424102278133772007at_nat ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__List__Olist_It__HOL__Obool_J,type,
    produc1275941116658105850t_bool: ( list_VEBT_VEBT * list_bool ) > produc7965489016306110858t_bool ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__List__Olist_It__Int__Oint_J,type,
    produc1392282695434103839st_int: ( list_VEBT_VEBT * list_int ) > produc7831203938951381541st_int ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc5570133714943300547st_nat: ( list_VEBT_VEBT * list_nat ) > produc1097915047028332489st_nat ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    produc3897820843166775703T_VEBT: ( list_VEBT_VEBT * list_VEBT_VEBT ) > produc9211091688327510695T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__HOL__Obool,type,
    produc3913204875133826030t_bool: ( nat * bool ) > produc8000372497851584572t_bool ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
    product_Pair_nat_int: ( nat * int ) > product_prod_nat_int ).

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

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

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc599794634098209291T_VEBT: ( nat * vEBT_VEBT ) > produc8025551001238799321T_VEBT ).

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

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

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

tff(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc6161850002892822231at_nat: ( product_prod_nat_nat * product_prod_nat_nat ) > produc859450856879609959at_nat ).

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Complex__Ocomplex,type,
    produc1693001998875562995omplex: ( real * complex ) > produc6979889472282505531omplex ).

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__HOL__Obool,type,
    produc3350604258665657362l_bool: ( real * bool ) > produc5468465692305236192l_bool ).

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Int__Oint,type,
    produc3179012173361985393al_int: ( real * int ) > produc8786904178792722361al_int ).

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Nat__Onat,type,
    produc3181502643871035669al_nat: ( real * nat ) > produc3741383161447143261al_nat ).

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
    produc4511245868158468465l_real: ( real * real ) > produc2422161461964618553l_real ).

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    produc6931449550656315951T_VEBT: ( real * vEBT_VEBT ) > produc3757001726724277373T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc4532415448927165861et_nat: ( set_nat * set_nat ) > produc7819656566062154093et_nat ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc2922128104949294807at_nat: ( set_Pr1261947904930325089at_nat * set_Pr1261947904930325089at_nat ) > produc3843707927480180839at_nat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Complex__Ocomplex,type,
    produc5617778602380981643omplex: ( vEBT_VEBT * complex ) > produc8380087813684007313omplex ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat,type,
    produc581526299967858633d_enat: ( vEBT_VEBT * extended_enat ) > produc7272778201969148633d_enat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
    produc3035007008238225530T_bool: ( vEBT_VEBT * bool ) > produc6448083994118892042T_bool ).

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

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

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    produc8117437818029410057T_real: ( vEBT_VEBT * real ) > produc5170161368751668367T_real ).

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

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc2004789812759622991on_nat: ( fun_nat_fun_nat_bool * produc4953844613479565601on_nat ) > produc3700678068921409119on_nat ).

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

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

tff(sy_c_Product__Type_OPair_001t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_Mt__HOL__Obool_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    produc8433281988922931063on_num: ( fun_num_fun_num_bool * produc3447558737645232053on_num ) > produc6167219322466932359on_num ).

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

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__HOL__Obool_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc5961576194287291153at_nat: ( fun_Pr248683899324676269t_bool * produc6121120109295599847at_nat ) > produc533602882674032673at_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc7814092408141220007at_nat: ( fun_Pr2477239776880934807at_nat * produc6121120109295599847at_nat ) > produc1260151622200656695at_nat ).

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

tff(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc6499014454317279255nteger: fun_fu4800041728187187127nteger ).

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

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

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__HOL__Obool,type,
    produc8560705615207772305t_bool: fun_int_fun_int_bool > fun_Pr803113274115125473t_bool ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc8211389475949308722nt_int: ( fun_int_fun_int_int * product_prod_int_int ) > int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4245557441103728435nt_int: ( fun_in1696471567231710611nt_int * product_prod_int_int ) > product_prod_int_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__HOL__Obool,type,
    produc8955392012077580105t_bool: fun_nat_fun_nat_bool > fun_Pr7757163693017731481t_bool ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    produc2761476792215241774st_nat: ( fun_na819858167508000762st_nat * product_prod_nat_nat ) > list_nat ).

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

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

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__HOL__Obool_J,type,
    produc5023667602222594023t_bool: fun_na4786964484885125043t_bool > fun_Pr248683899324676269t_bool ).

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

tff(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
    product_fst_int_int: product_prod_int_int > int ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
    product_snd_int_int: product_prod_int_int > int ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Oscomp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    produc5538323210962509403atural: ( fun_Pr7769407664385210434atural * fun_Co1105491363547889651atural ) > fun_Pr7769407664385210434atural ).

tff(sy_c_Random_Oiterate_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    iterat8892046348760725948atural: ( code_natural * fun_Co1105491363547889651atural * code_natural ) > fun_Pr7769407664385210434atural ).

tff(sy_c_Random_Olog,type,
    log: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Ominus__shift,type,
    minus_shift: ( code_natural * code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Onext,type,
    next: fun_Pr7769407664385210434atural ).

tff(sy_c_Random_Orange,type,
    range: code_natural > fun_Pr7769407664385210434atural ).

tff(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: fun_Pr1652463973008025892nt_rat ).

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

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun_ra4503298312962071866nt_int ).

tff(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
    field_5140801741446780682s_real: set_real ).

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

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun_Pr3145601080792579042t_bool ).

tff(sy_c_Rat_Opositive,type,
    positive: fun_rat_bool ).

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

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun_Pr3099820731930151597t_bool ).

tff(sy_c_Real_ORatreal,type,
    ratreal: fun_rat_real ).

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

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

tff(sy_c_Real_Opcr__real,type,
    pcr_real: fun_fu990398353789569848l_bool ).

tff(sy_c_Real_Opositive,type,
    positive2: fun_real_bool ).

tff(sy_c_Real_Orealrel,type,
    realrel: fun_fu2636726645394866687t_bool ).

tff(sy_c_Real_Orep__real,type,
    rep_real: fun_real_fun_nat_rat ).

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

tff(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
    real_V2521375963428798218omplex: set_complex ).

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

tff(sy_c_Real__Vector__Spaces_Olinear_001t__Real__Oreal_001t__Real__Oreal,type,
    real_V4572627801940501177l_real: fun_real_real > $o ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
    real_V1022390504157884413omplex: fun_complex_real ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
    real_V7735802525324610683m_real: fun_real_real ).

tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
    real_V4546457046886955230omplex: fun_real_complex ).

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

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

tff(sy_c_Relation_OField_001t__Nat__Onat,type,
    field_nat: set_Pr1261947904930325089at_nat > set_nat ).

tff(sy_c_Relation_OId_001t__Nat__Onat,type,
    id_nat2: set_Pr1261947904930325089at_nat ).

tff(sy_c_Relation_Oantisym_001t__Nat__Onat,type,
    antisym_nat: set_Pr1261947904930325089at_nat > $o ).

tff(sy_c_Relation_Orefl__on_001t__Nat__Onat,type,
    refl_on_nat: ( set_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_Relation_Ototal__on_001t__Nat__Onat,type,
    total_on_nat: ( set_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_Relation_Otrans_001t__Nat__Onat,type,
    trans_nat: set_Pr1261947904930325089at_nat > $o ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
    algebr932160517623751201me_int: fun_int_fun_int_bool ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
    algebr934650988132801477me_nat: fun_nat_fun_nat_bool ).

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

tff(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Onatural,type,
    divide5121882707175180666atural: ( code_natural * code_natural ) > code_natural ).

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

tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: fun_int_fun_int_int ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
    divide_divide_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
    divide_divide_rat: fun_rat_fun_rat_rat ).

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

tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
    dvd_dvd_Code_integer: fun_Co8310303586393106575r_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Onatural,type,
    dvd_dvd_Code_natural: ( code_natural * code_natural ) > $o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
    dvd_dvd_complex: fun_co1378794544173401837x_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: fun_int_fun_int_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
    dvd_dvd_rat: fun_rat_fun_rat_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
    dvd_dvd_real: fun_re1255401033344146673l_bool ).

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

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Onatural,type,
    modulo8411746178871703098atural: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: fun_int_fun_int_int ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
    modulo_modulo_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Ounit__factor__class_Ounit__factor_001t__Nat__Onat,type,
    unit_f2748546683901255202or_nat: fun_nat_nat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
    zero_n356916108424825756nteger: bool > code_integer ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Onatural,type,
    zero_n8403883297036319079atural: bool > code_natural ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
    zero_n1201886186963655149omplex: bool > complex ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: fun_bool_int ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
    zero_n2687167440665602831ol_nat: bool > nat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
    zero_n2052037380579107095ol_rat: bool > rat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
    zero_n3304061248610475627l_real: bool > real ).

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

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

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

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

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

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

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

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

tff(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
    collect_list_complex: fun_li356976171784545970x_bool > set_list_complex ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__HOL__Obool_J,type,
    collect_list_bool: fun_list_bool_bool > set_list_bool ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
    collect_list_int: fun_list_int_bool > set_list_int ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    collec5989764272469232197st_nat: fun_li3644157646236243712t_bool > set_list_list_nat ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
    collect_list_nat: fun_list_nat_bool > set_list_nat ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    collect_list_set_nat: fun_li4370653589787090394t_bool > set_list_set_nat ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    collec5608196760682091941T_VEBT: fun_li6006399982103069984T_bool > set_list_VEBT_VEBT ).

tff(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: fun_nat_bool > set_nat ).

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

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    collec3392354462482085612at_nat: fun_Pr7757163693017731481t_bool > set_Pr1261947904930325089at_nat ).

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

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

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

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

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

tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
    image_int_nat: ( fun_int_nat * set_int ) > set_nat ).

tff(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    image_list_nat_nat: ( fun_list_nat_nat * set_list_nat ) > set_nat ).

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

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    image_nat_list_nat: ( fun_nat_list_nat * set_nat ) > set_list_nat ).

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

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    image_5846123807819985514at_nat: ( fun_na1476630412504255114at_nat * set_nat ) > set_Pr1261947904930325089at_nat ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    image_nat_set_nat: ( fun_nat_set_nat * set_nat ) > set_set_nat ).

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

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
    image_678696785212003926at_nat: ( fun_na3635074868517587788at_nat * set_nat ) > set_Sum_sum_nat_nat ).

tff(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    image_2486076414777270412at_nat: ( fun_Pr2602117396471556196at_nat * set_Pr1261947904930325089at_nat ) > set_nat ).

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

tff(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
    image_char_nat: ( fun_char_nat * set_char ) > set_nat ).

tff(sy_c_Set_Oimage_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    image_1320371278474632150at_nat: ( fun_Su884838341992399436at_nat * set_Sum_sum_nat_nat ) > set_nat ).

tff(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
    insert_complex: ( complex * set_complex ) > set_complex ).

tff(sy_c_Set_Oinsert_001t__HOL__Obool,type,
    insert_bool: ( bool * set_bool ) > set_bool ).

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

tff(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
    insert_list_nat: ( list_nat * set_list_nat ) > set_list_nat ).

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

tff(sy_c_Set_Oinsert_001t__Num__Onum,type,
    insert_num: ( num * set_num ) > set_num ).

tff(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    insert8211810215607154385at_nat: ( product_prod_nat_nat * set_Pr1261947904930325089at_nat ) > set_Pr1261947904930325089at_nat ).

tff(sy_c_Set_Oinsert_001t__Rat__Orat,type,
    insert_rat: ( rat * set_rat ) > set_rat ).

tff(sy_c_Set_Oinsert_001t__Real__Oreal,type,
    insert_real: ( real * set_real ) > set_real ).

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

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

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

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Code____Numeral__Ointeger,type,
    set_fo1084959871951514735nteger: ( fun_na7078387940644931716nteger * nat * nat * code_integer ) > code_integer ).

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

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

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

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

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

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
    set_or1266510415728281911st_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
    set_or1269000886237332187st_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
    set_or7049704709247886629st_num: ( num * num ) > set_num ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
    set_or633870826150836451st_rat: ( rat * rat ) > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
    set_or1222579329274155063t_real: ( real * real ) > set_real ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4548717258645045905et_nat: ( set_nat * set_nat ) > set_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
    set_or4662586982721622107an_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
    set_ord_atMost_int: int > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
    set_ord_atMost_num: num > set_num ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
    set_ord_atMost_rat: rat > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
    set_ord_atMost_real: real > set_real ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4236626031148496127et_nat: set_nat > set_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
    set_or6656581121297822940st_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
    set_or6659071591806873216st_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
    set_or5832277885323065728an_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
    set_or5834768355832116004an_nat: nat > fun_nat_set_nat ).

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

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: fun_nat_set_nat ).

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

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

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: fun_nat_set_nat ).

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

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

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

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

tff(sy_c_String_Ochar_OChar,type,
    char2: ( bool * bool * bool * bool * bool * bool * bool * bool ) > char ).

tff(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

tff(sy_c_String_Ointeger__of__char,type,
    integer_of_char: char > code_integer ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

tff(sy_c_Sum__Type_OInl_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_Inl_nat_nat: nat > sum_sum_nat_nat ).

tff(sy_c_Sum__Type_OInr_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_Inr_nat_nat: nat > sum_sum_nat_nat ).

tff(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
    sum_ca7763040182479039464nt_nat: ( fun_nat_int * fun_nat_int * sum_sum_nat_nat ) > int ).

tff(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_ca6763686470577984908at_nat: ( fun_nat_nat * fun_nat_nat * sum_sum_nat_nat ) > nat ).

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

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

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

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

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

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

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

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log2: real > fun_real_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

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

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

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

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

tff(sy_c_Transitive__Closure_Otranclp_001t__Nat__Onat,type,
    transi2163837189807498211lp_nat: fun_nat_fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_Typerep_Otyperep_OTyperep,type,
    typerep2: ( literal * list_typerep ) > typerep ).

tff(sy_c_Typerep_Otyperep_Osize__typerep,type,
    size_typerep: fun_typerep_nat ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( bool * bool ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option4927543243414619207at_nat * nat * list_VEBT_VEBT * vEBT_VEBT ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
    vEBT_VEBT_elim_dead: ( vEBT_VEBT * extended_enat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
    vEBT_V312737461966249ad_rel: fun_Pr5442483421567724333t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list_VEBT_VEBT ) > fun_nat_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: vEBT_VEBT > fun_nat_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr817965127378462947t_bool ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr817965127378462947t_bool ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr817965127378462947t_bool ).

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

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set_nat ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_bool ).

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,
    vEBT_vebt_delete_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H,type,
    vEBT_VEBT_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H__rel,type,
    vEBT_VEBT_insert_rel: fun_Pr817965127378462947t_bool ).

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,
    vEBT_VEBT_bit_concat: nat > fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_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: fun_op7792301606653096457t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: fun_op7792301606653096457t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: fun_op7792301606653096457t_bool ).

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__comp__shift_001t__Nat__Onat,type,
    vEBT_V2881884560877996034ft_nat: fun_nat_fun_nat_bool > fun_op7792301606653096457t_bool ).

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

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

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun_op7210469962420818650on_nat ).

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

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun_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,
    vEBT_vebt_mint_rel: fun_VE5877677516038742509T_bool ).

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

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

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: set_nat > fun_nat_fun_nat_bool ).

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 ).

tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: fun_nat_fun_nat_bool > fun_nat_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: fun_Pr3099820731930151597t_bool > fun_Pr803113274115125473t_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: fun_Pr248683899324676269t_bool > fun_Pr7757163693017731481t_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
    accp_P6183159247885693666d_enat: ( fun_Pr5442483421567724333t_bool * produc7272778201969148633d_enat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: ( fun_Pr817965127378462947t_bool * produc9072475918466114483BT_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
    accp_VEBT_VEBT: fun_VE5877677516038742509T_bool > fun_VEBT_VEBT_bool ).

tff(sy_c_Wellfounded_Oless__than,type,
    less_than: set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

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

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__HOL__Obool,type,
    aa_Code_integer_bool: ( fun_Co8822979462182099027r_bool * code_integer ) > bool ).

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

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

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

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

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Rat__Orat,type,
    aa_Code_integer_rat: ( fun_Code_integer_rat * code_integer ) > rat ).

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

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__HOL__Obool_J,type,
    aa_Cod7250484176784686200r_bool: ( fun_Co8310303586393106575r_bool * code_integer ) > fun_Co8822979462182099027r_bool ).

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

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

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

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

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

tff(sy_c_aa_001t__Code____Numeral__Onatural_001t__Code____Numeral__Onatural,type,
    aa_Cod4272041906116450446atural: ( fun_Co7398788559141407565atural * code_natural ) > code_natural ).

tff(sy_c_aa_001t__Code____Numeral__Onatural_001t__Nat__Onat,type,
    aa_Code_natural_nat: ( fun_Code_natural_nat * code_natural ) > nat ).

tff(sy_c_aa_001t__Code____Numeral__Onatural_001t__fun_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    aa_Cod5141017464274028979atural: ( fun_Co6813818151677833714atural * code_natural ) > fun_Co7398788559141407565atural ).

tff(sy_c_aa_001t__Code____Numeral__Onatural_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J,type,
    aa_Cod6352003912080677724atural: ( fun_Co1105491363547889651atural * code_natural ) > fun_Pr7769407664385210434atural ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Code____Numeral__Ointeger,type,
    aa_com4217951221296830281nteger: ( fun_co8153928105989009288nteger * complex ) > code_integer ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    aa_complex_complex: ( fun_complex_complex * complex ) > complex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__HOL__Obool,type,
    aa_complex_bool: ( fun_complex_bool * complex ) > bool ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Int__Oint,type,
    aa_complex_int: ( fun_complex_int * complex ) > int ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    aa_complex_nat: ( fun_complex_nat * complex ) > nat ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Num__Onum,type,
    aa_complex_num: ( fun_complex_num * complex ) > num ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Rat__Orat,type,
    aa_complex_rat: ( fun_complex_rat * complex ) > rat ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    aa_complex_real: ( fun_complex_real * complex ) > real ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    aa_com3342990543769683525omplex: ( fun_co7673397065115134084omplex * complex ) > fun_complex_complex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Complex__Ocomplex_Mt__HOL__Obool_J,type,
    aa_com1553036987509291606x_bool: ( fun_co1378794544173401837x_bool * complex ) > fun_complex_bool ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Complex__Ocomplex_Mt__fun_It__Complex__Ocomplex_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J_J,type,
    aa_com177695874307951165omplex: ( fun_co3732688428756473980omplex * complex ) > fun_co6147893885205795665omplex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Complex__Ocomplex_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
    aa_com8177640530696437650omplex: ( fun_co6147893885205795665omplex * complex ) > fun_co3258106519541737638omplex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Complex__Ocomplex_Mt__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J_J,type,
    aa_com7713930848417683039omplex: ( fun_co3194742168270691230omplex * complex ) > fun_co6883864892131739507omplex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Int__Oint_Mt__Complex__Ocomplex_J,type,
    aa_com9156536935073639875omplex: ( fun_co7673900812888889602omplex * complex ) > fun_int_complex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
    aa_com2211550117870696024t_bool: ( fun_co2381122952212249071t_bool * complex ) > fun_int_bool ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
    aa_com8181001469867462503omplex: ( fun_co3258106519541737638omplex * complex ) > fun_nat_complex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
    aa_com5335841084339320244t_bool: ( fun_co2841405903858895691t_bool * complex ) > fun_nat_bool ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
    aa_com4347742694490471604omplex: ( fun_co6883864892131739507omplex * complex ) > fun_na1142814060743658952omplex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
    aa_com4766338864777664963omplex: ( fun_co286204877037288194omplex * complex ) > fun_real_complex ).

tff(sy_c_aa_001t__Complex__Ocomplex_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
    aa_com8943241093633526360l_bool: ( fun_co2286642472385812463l_bool * complex ) > fun_real_bool ).

tff(sy_c_aa_001t__Extended____Nat__Oenat_001t__fun_It__Nat__Onat_Mt__Extended____Nat__Oenat_J,type,
    aa_Ext1337733060012393797d_enat: ( fun_Ex2441982661509597004d_enat * extended_enat ) > fun_na8797796813480665199d_enat ).

tff(sy_c_aa_001t__HOL__Obool_001t__HOL__Obool,type,
    aa_bool_bool: ( fun_bool_bool * bool ) > bool ).

tff(sy_c_aa_001t__HOL__Obool_001t__Int__Oint,type,
    aa_bool_int: ( fun_bool_int * bool ) > int ).

tff(sy_c_aa_001t__HOL__Obool_001t__Nat__Onat,type,
    aa_bool_nat: ( fun_bool_nat * bool ) > nat ).

tff(sy_c_aa_001t__HOL__Obool_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J,type,
    aa_boo8970175072276269406l_bool: ( fun_bo1841576699660343469l_bool * bool ) > fun_bool_bool ).

tff(sy_c_aa_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    aa_int_Code_integer: ( fun_int_Code_integer * int ) > code_integer ).

tff(sy_c_aa_001t__Int__Oint_001t__Complex__Ocomplex,type,
    aa_int_complex: ( fun_int_complex * int ) > complex ).

tff(sy_c_aa_001t__Int__Oint_001t__HOL__Obool,type,
    aa_int_bool: ( fun_int_bool * int ) > bool ).

tff(sy_c_aa_001t__Int__Oint_001t__Int__Oint,type,
    aa_int_int: ( fun_int_int * int ) > int ).

tff(sy_c_aa_001t__Int__Oint_001t__Nat__Onat,type,
    aa_int_nat: ( fun_int_nat * int ) > nat ).

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tff(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Complex__Ocomplex_J,type,
    member8811922270175639012omplex: ( produc5838698208256999739omplex * set_Pr1846070511934368667omplex ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: ( product_prod_int_int * set_Pr958786334691620121nt_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
    member2744130022092475746t_real: ( produc679980390762269497t_real * set_Pr3538720872664544793t_real ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Code____Numeral__Ointeger_J_Mt__List__Olist_It__Code____Numeral__Ointeger_J_J,type,
    member749217712838834276nteger: ( produc862207588354017979nteger * set_Pr7565137564259432987nteger ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__HOL__Obool_J_Mt__List__Olist_It__HOL__Obool_J_J,type,
    member6300827366518646928t_bool: ( produc7619963554364372839t_bool * set_Pr4016434106243378759t_bool ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__HOL__Obool_J_Mt__List__Olist_It__Int__Oint_J_J,type,
    member2090666763348756895st_int: ( produc1591970419219485960st_int * set_Pr7659629972148914366st_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__HOL__Obool_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member4580749908280483651st_nat: ( produc4082053564151212716st_nat * set_Pr3249806354082057314st_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__HOL__Obool_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    member6981531815617313197T_VEBT: ( produc8209973772417936516T_VEBT * set_Pr3294270863102529764T_VEBT ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
    member6698963635872716290st_int: ( produc1186641810826059865st_int * set_Pr765067013931698361st_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__HOL__Obool_J_J,type,
    member702678106789554249t_bool: ( produc203981762660283314t_bool * set_Pr1340933433341948520t_bool ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member7340969449405702474st_nat: ( produc1828647624359046049st_nat * set_Pr3451248702717554689st_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    member5968030670617646438T_VEBT: ( produc872621073311890639T_VEBT * set_Pr1262583345697558789T_VEBT ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member6693912407220327184at_nat: ( produc6392793444374437607at_nat * set_Pr1542805901266377927at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member4080735728053443344at_nat: ( produc424102278133772007at_nat * set_Pr4333006031979791559at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__HOL__Obool_J_J,type,
    member6737047059505487539t_bool: ( produc7965489016306110858t_bool * set_Pr5897181536329667306t_bool ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__Int__Oint_J_J,type,
    member3703241499402361532st_int: ( produc7831203938951381541st_int * set_Pr4080907618048478043st_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member6193324644334088288st_nat: ( produc1097915047028332489st_nat * set_Pr8894456036836396799st_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    member4439316823752958928T_VEBT: ( produc9211091688327510695T_VEBT * set_Pr1916528119006554503T_VEBT ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: ( product_prod_nat_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
    member8549952807677709168T_VEBT: ( produc8025551001238799321T_VEBT * set_Pr6167073792073659919T_VEBT ) > $o ).

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__Real__Oreal_Mt__Complex__Ocomplex_J,type,
    member7358116576843751780omplex: ( produc6979889472282505531omplex * set_Pr6591433984475009307omplex ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__HOL__Obool_J,type,
    member3295258257541327095l_bool: ( produc5468465692305236192l_bool * set_Pr2904644018331178518l_bool ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
    member1627681773268152802al_int: ( produc8786904178792722361al_int * set_Pr1019928272762051225al_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
    member5805532792777349510al_nat: ( produc3741383161447143261al_nat * set_Pr3510011417693777981al_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    member7849222048561428706l_real: ( produc2422161461964618553l_real * set_Pr6218003697084177305l_real ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
    member7262085504369356948T_VEBT: ( produc3757001726724277373T_VEBT * set_Pr6019664923565264691T_VEBT ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member8277197624267554838et_nat: ( produc7819656566062154093et_nat * set_Pr5488025237498180813et_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__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Complex__Ocomplex_J,type,
    member3207599676835851048omplex: ( produc8380087813684007313omplex * set_Pr216944050393469383omplex ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J,type,
    member3504487111852181555T_bool: ( produc6448083994118892042T_bool * set_Pr2229214029182756970T_bool ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    member5419026705395827622BT_int: ( produc4894624898956917775BT_int * set_Pr5066593544530342725BT_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    member373505688050248522BT_nat: ( produc9072475918466114483BT_nat * set_Pr7556676689462069481BT_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    member8675245146396747942T_real: ( produc5170161368751668367T_real * set_Pr7765410600122031685T_real ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    member568628332442017744T_VEBT: ( produc8243902056947475879T_VEBT * set_Pr6192946355708809607T_VEBT ) > $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 > fun_set_real_bool ).

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__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2643936169264416010at_nat: ( set_Pr1261947904930325089at_nat * set_se7855581050983116737at_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_a____,type,
    a: bool ).

tff(sy_v_b____,type,
    b: bool ).

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_info____,type,
    info: option4927543243414619207at_nat ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_sa____,type,
    sa: vEBT_VEBT ).

tff(sy_v_summary_H____,type,
    summary: vEBT_VEBT ).

tff(sy_v_summary____,type,
    summary2: vEBT_VEBT ).

tff(sy_v_treeList_H____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList2: list_VEBT_VEBT ).

% Relevant facts (10207)
tff(fact_0_case4_I9_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),ma)) ).

% case4(9)
tff(fact_1_option_Oinject,axiom,
    ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( some_P7363390416028606310at_nat(X2) = some_P7363390416028606310at_nat(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_2_option_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( some_nat(X2) = some_nat(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_3_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( some_num(X2) = some_num(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_4_prod_Oinject,axiom,
    ! [X1: code_integer,X2: code_integer,Y1: code_integer,Y2: code_integer] :
      ( ( produc1086072967326762835nteger(X1,X2) = produc1086072967326762835nteger(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_5_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X2: product_prod_nat_nat,Y1: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( produc6161850002892822231at_nat(X1,X2) = produc6161850002892822231at_nat(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_6_prod_Oinject,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,Y1: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
      ( ( produc2922128104949294807at_nat(X1,X2) = produc2922128104949294807at_nat(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_7_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_8_prod_Oinject,axiom,
    ! [X1: int,X2: int,Y1: int,Y2: int] :
      ( ( product_Pair_int_int(X1,X2) = product_Pair_int_int(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_9_old_Oprod_Oinject,axiom,
    ! [A: code_integer,B: code_integer,A2: code_integer,B2: code_integer] :
      ( ( produc1086072967326762835nteger(A,B) = produc1086072967326762835nteger(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_10_old_Oprod_Oinject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( produc6161850002892822231at_nat(A,B) = produc6161850002892822231at_nat(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_11_old_Oprod_Oinject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( produc2922128104949294807at_nat(A,B) = produc2922128104949294807at_nat(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_12_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_13_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( product_Pair_int_int(A,B) = product_Pair_int_int(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_14_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K: nat,M: nat] : X != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K),M) ).

% prod_decode_aux.cases
tff(fact_15_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8923325533196201883nteger] :
      ~ ! [A3: code_integer,B3: code_integer] : Y != produc1086072967326762835nteger(A3,B3) ).

% old.prod.exhaust
tff(fact_16_old_Oprod_Oexhaust,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] : Y != produc6161850002892822231at_nat(A3,B3) ).

% old.prod.exhaust
tff(fact_17_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3843707927480180839at_nat] :
      ~ ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : Y != produc2922128104949294807at_nat(A3,B3) ).

% old.prod.exhaust
tff(fact_18_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] : Y != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) ).

% old.prod.exhaust
tff(fact_19_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A3: int,B3: int] : Y != product_Pair_int_int(A3,B3) ).

% old.prod.exhaust
tff(fact_20_surj__pair,axiom,
    ! [P: produc8923325533196201883nteger] :
    ? [X3: code_integer,Y3: code_integer] : P = produc1086072967326762835nteger(X3,Y3) ).

% surj_pair
tff(fact_21_surj__pair,axiom,
    ! [P: produc859450856879609959at_nat] :
    ? [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] : P = produc6161850002892822231at_nat(X3,Y3) ).

% surj_pair
tff(fact_22_surj__pair,axiom,
    ! [P: produc3843707927480180839at_nat] :
    ? [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : P = produc2922128104949294807at_nat(X3,Y3) ).

% surj_pair
tff(fact_23_surj__pair,axiom,
    ! [P: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] : P = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3) ).

% surj_pair
tff(fact_24_surj__pair,axiom,
    ! [P: product_prod_int_int] :
    ? [X3: int,Y3: int] : P = product_Pair_int_int(X3,Y3) ).

% surj_pair
tff(fact_25_prod__cases,axiom,
    ! [P2: fun_Pr3195429412498512255r_bool,P: produc8923325533196201883nteger] :
      ( ! [A3: code_integer,B3: code_integer] : pp(aa_Pro5843519176478241656r_bool(P2,produc1086072967326762835nteger(A3,B3)))
     => pp(aa_Pro5843519176478241656r_bool(P2,P)) ) ).

% prod_cases
tff(fact_26_prod__cases,axiom,
    ! [P2: fun_Pr446604468264085555t_bool,P: produc859450856879609959at_nat] :
      ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] : pp(aa_Pro6098352494050694700t_bool(P2,produc6161850002892822231at_nat(A3,B3)))
     => pp(aa_Pro6098352494050694700t_bool(P2,P)) ) ).

% prod_cases
tff(fact_27_prod__cases,axiom,
    ! [P2: fun_Pr1607443439640623155t_bool,P: produc3843707927480180839at_nat] :
      ( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : pp(aa_Pro1809015098595223084t_bool(P2,produc2922128104949294807at_nat(A3,B3)))
     => pp(aa_Pro1809015098595223084t_bool(P2,P)) ) ).

% prod_cases
tff(fact_28_prod__cases,axiom,
    ! [P2: fun_Pr7757163693017731481t_bool,P: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : pp(aa_Pro2232559814647311250t_bool(P2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3)))
     => pp(aa_Pro2232559814647311250t_bool(P2,P)) ) ).

% prod_cases
tff(fact_29_prod__cases,axiom,
    ! [P2: fun_Pr803113274115125473t_bool,P: product_prod_int_int] :
      ( ! [A3: int,B3: int] : pp(aa_Pro4501881432599481050t_bool(P2,product_Pair_int_int(A3,B3)))
     => pp(aa_Pro4501881432599481050t_bool(P2,P)) ) ).

% prod_cases
tff(fact_30_Pair__inject,axiom,
    ! [A: code_integer,B: code_integer,A2: code_integer,B2: code_integer] :
      ( ( produc1086072967326762835nteger(A,B) = produc1086072967326762835nteger(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_31_Pair__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( produc6161850002892822231at_nat(A,B) = produc6161850002892822231at_nat(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_32_Pair__inject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( produc2922128104949294807at_nat(A,B) = produc2922128104949294807at_nat(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_33_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_34_Pair__inject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( product_Pair_int_int(A,B) = product_Pair_int_int(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_35_prod__cases3,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A3: product_prod_nat_nat,B3: nat,C: nat] : Y != produc6161850002892822231at_nat(A3,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),C)) ).

% prod_cases3
tff(fact_36_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))
        & ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),Xs))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),X)) ) ) ) ).

% max_in_set_def
tff(fact_37_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))
        & ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),Xs))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),X4)) ) ) ) ).

% min_in_set_def
tff(fact_38_lesseq__shift,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,some_nat(X)),some_nat(Y))) ) ).

% lesseq_shift
tff(fact_39_prod__induct3,axiom,
    ! [P2: fun_Pr446604468264085555t_bool,X: produc859450856879609959at_nat] :
      ( ! [A3: product_prod_nat_nat,B3: nat,C: nat] : pp(aa_Pro6098352494050694700t_bool(P2,produc6161850002892822231at_nat(A3,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),C))))
     => pp(aa_Pro6098352494050694700t_bool(P2,X)) ) ).

% prod_induct3
tff(fact_40_order__refl,axiom,
    ! [X: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(X),X)) ).

% order_refl
tff(fact_41_order__refl,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),X)) ).

% order_refl
tff(fact_42_order__refl,axiom,
    ! [X: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),X)) ).

% order_refl
tff(fact_43_order__refl,axiom,
    ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),X)) ).

% order_refl
tff(fact_44_order__refl,axiom,
    ! [X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),X)) ).

% order_refl
tff(fact_45_dual__order_Orefl,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(A),A)) ).

% dual_order.refl
tff(fact_46_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_47_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_48_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_49_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_50_relChain__def,axiom,
    ! [R: set_Pr4811707699266497531nteger,As: fun_Code_integer_rat] :
      ( bNF_Ca8354645632395198811er_rat(R,As)
    <=> ! [I: code_integer,J: code_integer] :
          ( member157494554546826820nteger(produc1086072967326762835nteger(I,J),R)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Code_integer_rat(As,I)),aa_Code_integer_rat(As,J))) ) ) ).

% relChain_def
tff(fact_51_relChain__def,axiom,
    ! [R: set_Pr1261947904930325089at_nat,As: fun_nat_rat] :
      ( bNF_Ca333620267926924494at_rat(R,As)
    <=> ! [I: nat,J: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(I),J),R)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(As,I)),aa_nat_rat(As,J))) ) ) ).

% relChain_def
tff(fact_52_relChain__def,axiom,
    ! [R: set_Pr958786334691620121nt_int,As: fun_int_rat] :
      ( bNF_Ca1332973979827979050nt_rat(R,As)
    <=> ! [I: int,J: int] :
          ( member5262025264175285858nt_int(product_Pair_int_int(I,J),R)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(As,I)),aa_int_rat(As,J))) ) ) ).

% relChain_def
tff(fact_53_relChain__def,axiom,
    ! [R: set_Pr4811707699266497531nteger,As: fun_Code_integer_num] :
      ( bNF_Ca5547107478637473181er_num(R,As)
    <=> ! [I: code_integer,J: code_integer] :
          ( member157494554546826820nteger(produc1086072967326762835nteger(I,J),R)
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_Code_integer_num(As,I)),aa_Code_integer_num(As,J))) ) ) ).

% relChain_def
tff(fact_54_relChain__def,axiom,
    ! [R: set_Pr1261947904930325089at_nat,As: fun_nat_num] :
      ( bNF_Ca6749454151023974672at_num(R,As)
    <=> ! [I: nat,J: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(I),J),R)
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(As,I)),aa_nat_num(As,J))) ) ) ).

% relChain_def
tff(fact_55_relChain__def,axiom,
    ! [R: set_Pr958786334691620121nt_int,As: fun_int_num] :
      ( bNF_Ca7748807862925029228nt_num(R,As)
    <=> ! [I: int,J: int] :
          ( member5262025264175285858nt_int(product_Pair_int_int(I,J),R)
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(As,I)),aa_int_num(As,J))) ) ) ).

% relChain_def
tff(fact_56_relChain__def,axiom,
    ! [R: set_Pr4811707699266497531nteger,As: fun_Code_integer_nat] :
      ( bNF_Ca8989775692481694547er_nat(R,As)
    <=> ! [I: code_integer,J: code_integer] :
          ( member157494554546826820nteger(produc1086072967326762835nteger(I,J),R)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Code_integer_nat(As,I)),aa_Code_integer_nat(As,J))) ) ) ).

% relChain_def
tff(fact_57_relChain__def,axiom,
    ! [R: set_Pr1261947904930325089at_nat,As: fun_nat_nat] :
      ( bNF_Ca968750328013420230at_nat(R,As)
    <=> ! [I: nat,J: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(I),J),R)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(As,I)),aa_nat_nat(As,J))) ) ) ).

% relChain_def
tff(fact_58_relChain__def,axiom,
    ! [R: set_Pr958786334691620121nt_int,As: fun_int_nat] :
      ( bNF_Ca1968104039914474786nt_nat(R,As)
    <=> ! [I: int,J: int] :
          ( member5262025264175285858nt_int(product_Pair_int_int(I,J),R)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(As,I)),aa_int_nat(As,J))) ) ) ).

% relChain_def
tff(fact_59_relChain__def,axiom,
    ! [R: set_Pr4811707699266497531nteger,As: fun_Code_integer_int] :
      ( bNF_Ca8987285221972644271er_int(R,As)
    <=> ! [I: code_integer,J: code_integer] :
          ( member157494554546826820nteger(produc1086072967326762835nteger(I,J),R)
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_Code_integer_int(As,I)),aa_Code_integer_int(As,J))) ) ) ).

% relChain_def
tff(fact_60_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc9217583074051984497at_nat] :
      ~ ! [F: fun_nat_fun_nat_nat,A3: nat,B3: nat,Acc: nat] : X != produc2818902965508115497at_nat(F,produc487386426758144856at_nat(A3,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_61_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_62_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_63_lift__Suc__mono__le,axiom,
    ! [F2: fun_nat_set_nat,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F2,N3)),aa_nat_set_nat(F2,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F2,N)),aa_nat_set_nat(F2,N2))) ) ) ).

% lift_Suc_mono_le
tff(fact_64_lift__Suc__mono__le,axiom,
    ! [F2: fun_nat_rat,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,N3)),aa_nat_rat(F2,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,N)),aa_nat_rat(F2,N2))) ) ) ).

% lift_Suc_mono_le
tff(fact_65_lift__Suc__mono__le,axiom,
    ! [F2: fun_nat_num,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F2,N3)),aa_nat_num(F2,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F2,N)),aa_nat_num(F2,N2))) ) ) ).

% lift_Suc_mono_le
tff(fact_66_lift__Suc__mono__le,axiom,
    ! [F2: fun_nat_nat,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F2,N3)),aa_nat_nat(F2,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F2,N)),aa_nat_nat(F2,N2))) ) ) ).

% lift_Suc_mono_le
tff(fact_67_lift__Suc__mono__le,axiom,
    ! [F2: fun_nat_int,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F2,N3)),aa_nat_int(F2,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F2,N)),aa_nat_int(F2,N2))) ) ) ).

% lift_Suc_mono_le
tff(fact_68_lift__Suc__antimono__le,axiom,
    ! [F2: fun_nat_set_nat,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F2,aa_nat_nat(suc,N3))),aa_nat_set_nat(F2,N3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F2,N2)),aa_nat_set_nat(F2,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_69_lift__Suc__antimono__le,axiom,
    ! [F2: fun_nat_rat,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,aa_nat_nat(suc,N3))),aa_nat_rat(F2,N3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,N2)),aa_nat_rat(F2,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_70_lift__Suc__antimono__le,axiom,
    ! [F2: fun_nat_num,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F2,aa_nat_nat(suc,N3))),aa_nat_num(F2,N3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F2,N2)),aa_nat_num(F2,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_71_lift__Suc__antimono__le,axiom,
    ! [F2: fun_nat_nat,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F2,aa_nat_nat(suc,N3))),aa_nat_nat(F2,N3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F2,N2)),aa_nat_nat(F2,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_72_lift__Suc__antimono__le,axiom,
    ! [F2: fun_nat_int,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F2,aa_nat_nat(suc,N3))),aa_nat_int(F2,N3)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F2,N2)),aa_nat_int(F2,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_73_le__refl,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N)) ).

% le_refl
tff(fact_74_le__trans,axiom,
    ! [I2: nat,J2: nat,K2: 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_eq_nat,J2),K2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),K2)) ) ) ).

% le_trans
tff(fact_75__C1_C_I2_J,axiom,
    deg = aa_nat_nat(suc,zero_zero_nat) ).

% "1"(2)
tff(fact_76_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_77_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_78_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_79_Suc__le__mono,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_nat_nat(suc,M2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ).

% Suc_le_mono
tff(fact_80_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa_nat_nat(suc,N) ).

% n_not_Suc_n
tff(fact_81_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_82_transitive__stepwise__le,axiom,
    ! [M2: nat,N: nat,R2: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( ! [X3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X3),X3))
       => ( ! [X3: nat,Y3: nat,Z: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X3),Y3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,Y3),Z))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X3),Z)) ) )
         => ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,N3),aa_nat_nat(suc,N3)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,M2),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_83_nat__induct__at__least,axiom,
    ! [M2: nat,N: nat,P2: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( pp(aa_nat_bool(P2,M2))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N3))
             => ( pp(aa_nat_bool(P2,N3))
               => pp(aa_nat_bool(P2,aa_nat_nat(suc,N3))) ) )
         => pp(aa_nat_bool(P2,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_84_mem__Collect__eq,axiom,
    ! [A: int,P2: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(A),collect_int(P2)))
    <=> pp(aa_int_bool(P2,A)) ) ).

% mem_Collect_eq
tff(fact_85_mem__Collect__eq,axiom,
    ! [A: complex,P2: fun_complex_bool] :
      ( pp(aa_set_complex_bool(member_complex(A),collect_complex(P2)))
    <=> pp(aa_complex_bool(P2,A)) ) ).

% mem_Collect_eq
tff(fact_86_mem__Collect__eq,axiom,
    ! [A: real,P2: fun_real_bool] :
      ( pp(aa_set_real_bool(member_real(A),collect_real(P2)))
    <=> pp(aa_real_bool(P2,A)) ) ).

% mem_Collect_eq
tff(fact_87_mem__Collect__eq,axiom,
    ! [A: list_nat,P2: fun_list_nat_bool] :
      ( member_list_nat(A,collect_list_nat(P2))
    <=> pp(aa_list_nat_bool(P2,A)) ) ).

% mem_Collect_eq
tff(fact_88_mem__Collect__eq,axiom,
    ! [A: set_nat,P2: fun_set_nat_bool] :
      ( member_set_nat(A,collect_set_nat(P2))
    <=> pp(aa_set_nat_bool(P2,A)) ) ).

% mem_Collect_eq
tff(fact_89_mem__Collect__eq,axiom,
    ! [A: nat,P2: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(A),collect_nat(P2)))
    <=> pp(aa_nat_bool(P2,A)) ) ).

% mem_Collect_eq
tff(fact_90_Collect__mem__eq,axiom,
    ! [A4: set_int] : collect_int(aTP_Lamm_a(A4)) = A4 ).

% Collect_mem_eq
tff(fact_91_Collect__mem__eq,axiom,
    ! [A4: set_complex] : collect_complex(aTP_Lamm_aa(A4)) = A4 ).

% Collect_mem_eq
tff(fact_92_Collect__mem__eq,axiom,
    ! [A4: set_real] : collect_real(aTP_Lamm_ab(A4)) = A4 ).

% Collect_mem_eq
tff(fact_93_Collect__mem__eq,axiom,
    ! [A4: set_list_nat] : collect_list_nat(aTP_Lamm_ac(A4)) = A4 ).

% Collect_mem_eq
tff(fact_94_Collect__mem__eq,axiom,
    ! [A4: set_set_nat] : collect_set_nat(aTP_Lamm_ad(A4)) = A4 ).

% Collect_mem_eq
tff(fact_95_Collect__mem__eq,axiom,
    ! [A4: set_nat] : collect_nat(aTP_Lamm_ae(A4)) = A4 ).

% Collect_mem_eq
tff(fact_96_Collect__cong,axiom,
    ! [P2: fun_complex_bool,Q: fun_complex_bool] :
      ( ! [X3: complex] :
          ( pp(aa_complex_bool(P2,X3))
        <=> pp(aa_complex_bool(Q,X3)) )
     => ( collect_complex(P2) = collect_complex(Q) ) ) ).

% Collect_cong
tff(fact_97_Collect__cong,axiom,
    ! [P2: fun_real_bool,Q: fun_real_bool] :
      ( ! [X3: real] :
          ( pp(aa_real_bool(P2,X3))
        <=> pp(aa_real_bool(Q,X3)) )
     => ( collect_real(P2) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_98_Collect__cong,axiom,
    ! [P2: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ! [X3: list_nat] :
          ( pp(aa_list_nat_bool(P2,X3))
        <=> pp(aa_list_nat_bool(Q,X3)) )
     => ( collect_list_nat(P2) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_99_Collect__cong,axiom,
    ! [P2: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ! [X3: set_nat] :
          ( pp(aa_set_nat_bool(P2,X3))
        <=> pp(aa_set_nat_bool(Q,X3)) )
     => ( collect_set_nat(P2) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_100_Collect__cong,axiom,
    ! [P2: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_nat_bool(P2,X3))
        <=> pp(aa_nat_bool(Q,X3)) )
     => ( collect_nat(P2) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_101_full__nat__induct,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M3)),N3))
             => pp(aa_nat_bool(P2,M3)) )
         => pp(aa_nat_bool(P2,N3)) )
     => pp(aa_nat_bool(P2,N)) ) ).

% full_nat_induct
tff(fact_102_not__less__eq__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M2)) ) ).

% not_less_eq_eq
tff(fact_103_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),N)) ).

% Suc_n_not_le_n
tff(fact_104_le__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
        | ( M2 = aa_nat_nat(suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_105_Suc__le__D,axiom,
    ! [N: nat,M4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M4))
     => ? [M: nat] : M4 = aa_nat_nat(suc,M) ) ).

% Suc_le_D
tff(fact_106_le__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N))) ) ).

% le_SucI
tff(fact_107_le__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( M2 = aa_nat_nat(suc,N) ) ) ) ).

% le_SucE
tff(fact_108_Suc__leD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% Suc_leD
tff(fact_109_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_110_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_111_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_112_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_113_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_114_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_115_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_116_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_117_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_118_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F2: fun_rat_rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_rat(F2,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F2,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_119_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F2: fun_rat_num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_num(F2,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F2,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_120_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F2: fun_rat_nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_nat(F2,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F2,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_121_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F2: fun_rat_int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_int(F2,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F2,X3)),aa_rat_int(F2,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F2,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_122_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F2: fun_num_rat,C2: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_rat(F2,B) = C2 )
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F2,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_123_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F2: fun_num_num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_num(F2,B) = C2 )
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F2,X3)),aa_num_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F2,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_124_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F2: fun_num_nat,C2: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_nat(F2,B) = C2 )
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F2,X3)),aa_num_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F2,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_125_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F2: fun_num_int,C2: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_int(F2,B) = C2 )
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F2,X3)),aa_num_int(F2,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F2,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_126_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F2: fun_nat_rat,C2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( aa_nat_rat(F2,B) = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,X3)),aa_nat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_127_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F2: fun_nat_num,C2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( aa_nat_num(F2,B) = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F2,X3)),aa_nat_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F2,A)),C2)) ) ) ) ).

% ord_le_eq_subst
tff(fact_128_ord__eq__le__subst,axiom,
    ! [A: rat,F2: fun_rat_rat,B: rat,C2: rat] :
      ( ( A = aa_rat_rat(F2,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F2,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_129_ord__eq__le__subst,axiom,
    ! [A: num,F2: fun_rat_num,B: rat,C2: rat] :
      ( ( A = aa_rat_num(F2,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F2,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_130_ord__eq__le__subst,axiom,
    ! [A: nat,F2: fun_rat_nat,B: rat,C2: rat] :
      ( ( A = aa_rat_nat(F2,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F2,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_131_ord__eq__le__subst,axiom,
    ! [A: int,F2: fun_rat_int,B: rat,C2: rat] :
      ( ( A = aa_rat_int(F2,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F2,X3)),aa_rat_int(F2,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_rat_int(F2,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_132_ord__eq__le__subst,axiom,
    ! [A: rat,F2: fun_num_rat,B: num,C2: num] :
      ( ( A = aa_num_rat(F2,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F2,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_133_ord__eq__le__subst,axiom,
    ! [A: num,F2: fun_num_num,B: num,C2: num] :
      ( ( A = aa_num_num(F2,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F2,X3)),aa_num_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F2,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_134_ord__eq__le__subst,axiom,
    ! [A: nat,F2: fun_num_nat,B: num,C2: num] :
      ( ( A = aa_num_nat(F2,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F2,X3)),aa_num_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F2,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_135_ord__eq__le__subst,axiom,
    ! [A: int,F2: fun_num_int,B: num,C2: num] :
      ( ( A = aa_num_int(F2,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F2,X3)),aa_num_int(F2,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_num_int(F2,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_136_ord__eq__le__subst,axiom,
    ! [A: rat,F2: fun_nat_rat,B: nat,C2: nat] :
      ( ( A = aa_nat_rat(F2,B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,X3)),aa_nat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F2,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_137_ord__eq__le__subst,axiom,
    ! [A: num,F2: fun_nat_num,B: nat,C2: nat] :
      ( ( A = aa_nat_num(F2,B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F2,X3)),aa_nat_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F2,C2))) ) ) ) ).

% ord_eq_le_subst
tff(fact_138_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_139_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_140_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_141_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_142_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_143_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_144_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_145_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_146_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_147_order__subst2,axiom,
    ! [A: rat,B: rat,F2: fun_rat_rat,C2: 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(F2,B)),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F2,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_148_order__subst2,axiom,
    ! [A: rat,B: rat,F2: fun_rat_num,C2: 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(F2,B)),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F2,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_149_order__subst2,axiom,
    ! [A: rat,B: rat,F2: fun_rat_nat,C2: 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(F2,B)),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F2,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_150_order__subst2,axiom,
    ! [A: rat,B: rat,F2: fun_rat_int,C2: 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(F2,B)),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F2,X3)),aa_rat_int(F2,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F2,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_151_order__subst2,axiom,
    ! [A: num,B: num,F2: fun_num_rat,C2: 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(F2,B)),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F2,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_152_order__subst2,axiom,
    ! [A: num,B: num,F2: fun_num_num,C2: 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(F2,B)),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F2,X3)),aa_num_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F2,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_153_order__subst2,axiom,
    ! [A: num,B: num,F2: fun_num_nat,C2: 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(F2,B)),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F2,X3)),aa_num_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F2,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_154_order__subst2,axiom,
    ! [A: num,B: num,F2: fun_num_int,C2: 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(F2,B)),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F2,X3)),aa_num_int(F2,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F2,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_155_order__subst2,axiom,
    ! [A: nat,B: nat,F2: fun_nat_rat,C2: 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(F2,B)),C2))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,X3)),aa_nat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_156_order__subst2,axiom,
    ! [A: nat,B: nat,F2: fun_nat_num,C2: 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(F2,B)),C2))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F2,X3)),aa_nat_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F2,A)),C2)) ) ) ) ).

% order_subst2
tff(fact_157_order__subst1,axiom,
    ! [A: rat,F2: fun_rat_rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F2,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F2,C2))) ) ) ) ).

% order_subst1
tff(fact_158_order__subst1,axiom,
    ! [A: rat,F2: fun_num_rat,B: num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F2,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F2,C2))) ) ) ) ).

% order_subst1
tff(fact_159_order__subst1,axiom,
    ! [A: rat,F2: fun_nat_rat,B: nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F2,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,X3)),aa_nat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F2,C2))) ) ) ) ).

% order_subst1
tff(fact_160_order__subst1,axiom,
    ! [A: rat,F2: fun_int_rat,B: int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F2,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C2))
       => ( ! [X3: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F2,X3)),aa_int_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F2,C2))) ) ) ) ).

% order_subst1
tff(fact_161_order__subst1,axiom,
    ! [A: num,F2: fun_rat_num,B: rat,C2: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F2,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F2,C2))) ) ) ) ).

% order_subst1
tff(fact_162_order__subst1,axiom,
    ! [A: num,F2: fun_num_num,B: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F2,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F2,X3)),aa_num_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F2,C2))) ) ) ) ).

% order_subst1
tff(fact_163_order__subst1,axiom,
    ! [A: num,F2: fun_nat_num,B: nat,C2: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F2,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F2,X3)),aa_nat_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F2,C2))) ) ) ) ).

% order_subst1
tff(fact_164_order__subst1,axiom,
    ! [A: num,F2: fun_int_num,B: int,C2: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_int_num(F2,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C2))
       => ( ! [X3: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F2,X3)),aa_int_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_int_num(F2,C2))) ) ) ) ).

% order_subst1
tff(fact_165_order__subst1,axiom,
    ! [A: nat,F2: fun_rat_nat,B: rat,C2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F2,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F2,C2))) ) ) ) ).

% order_subst1
tff(fact_166_order__subst1,axiom,
    ! [A: nat,F2: fun_num_nat,B: num,C2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F2,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F2,X3)),aa_num_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F2,C2))) ) ) ) ).

% order_subst1
tff(fact_167_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_168_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_169_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_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C2),B))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(C2),A)) ) ) ).

% dual_order.trans
tff(fact_178_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C2: 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,C2),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A)) ) ) ).

% dual_order.trans
tff(fact_179_dual__order_Otrans,axiom,
    ! [B: num,A: num,C2: 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,C2),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A)) ) ) ).

% dual_order.trans
tff(fact_180_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C2: 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,C2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A)) ) ) ).

% dual_order.trans
tff(fact_181_dual__order_Otrans,axiom,
    ! [B: int,A: int,C2: 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,C2),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A)) ) ) ).

% dual_order.trans
tff(fact_182_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_183_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_184_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_185_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_186_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_187_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_188_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_189_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_190_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_191_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_192_linorder__wlog,axiom,
    ! [P2: 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(P2,A3),B3)) )
     => ( ! [A3: rat,B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,B3),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,A3),B3)) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,A),B)) ) ) ).

% linorder_wlog
tff(fact_193_linorder__wlog,axiom,
    ! [P2: 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(P2,A3),B3)) )
     => ( ! [A3: num,B3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(P2,B3),A3))
           => pp(aa_num_bool(aa_num_fun_num_bool(P2,A3),B3)) )
       => pp(aa_num_bool(aa_num_fun_num_bool(P2,A),B)) ) ) ).

% linorder_wlog
tff(fact_194_linorder__wlog,axiom,
    ! [P2: 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(P2,A3),B3)) )
     => ( ! [A3: nat,B3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,B3),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A3),B3)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A),B)) ) ) ).

% linorder_wlog
tff(fact_195_linorder__wlog,axiom,
    ! [P2: 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(P2,A3),B3)) )
     => ( ! [A3: int,B3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(P2,B3),A3))
           => pp(aa_int_bool(aa_int_fun_int_bool(P2,A3),B3)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P2,A),B)) ) ) ).

% linorder_wlog
tff(fact_196_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_197_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_198_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_199_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_200_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_201_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C2: 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),C2))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C2)) ) ) ).

% order.trans
tff(fact_202_order_Otrans,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C2)) ) ) ).

% order.trans
tff(fact_203_order_Otrans,axiom,
    ! [A: num,B: num,C2: 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),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C2)) ) ) ).

% order.trans
tff(fact_204_order_Otrans,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C2)) ) ) ).

% order.trans
tff(fact_205_order_Otrans,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C2)) ) ) ).

% order.trans
tff(fact_206_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_207_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_208_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_209_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_210_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_211_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( ( B = C2 )
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C2)) ) ) ).

% ord_le_eq_trans
tff(fact_212_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( B = C2 )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C2)) ) ) ).

% ord_le_eq_trans
tff(fact_213_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( B = C2 )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C2)) ) ) ).

% ord_le_eq_trans
tff(fact_214_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( B = C2 )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C2)) ) ) ).

% ord_le_eq_trans
tff(fact_215_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( ( B = C2 )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C2)) ) ) ).

% ord_le_eq_trans
tff(fact_216_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( A = B )
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),C2))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C2)) ) ) ).

% ord_eq_le_trans
tff(fact_217_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C2)) ) ) ).

% ord_eq_le_trans
tff(fact_218_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( A = B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C2)) ) ) ).

% ord_eq_le_trans
tff(fact_219_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C2)) ) ) ).

% ord_eq_le_trans
tff(fact_220_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C2)) ) ) ).

% ord_eq_le_trans
tff(fact_221_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_222_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_223_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_224_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_225_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_226_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_227_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_228_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_229_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_230_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_231_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_232_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_233_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_234_bounded__Max__nat,axiom,
    ! [P2: fun_nat_bool,X: nat,M5: nat] :
      ( pp(aa_nat_bool(P2,X))
     => ( ! [X3: nat] :
            ( pp(aa_nat_bool(P2,X3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),M5)) )
       => ~ ! [M: nat] :
              ( pp(aa_nat_bool(P2,M))
             => ~ ! [X5: nat] :
                    ( pp(aa_nat_bool(P2,X5))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),M)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_235_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: fun_nat_bool,K2: nat,B: nat] :
      ( pp(aa_nat_bool(P2,K2))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P2,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B)) )
       => ? [X3: nat] :
            ( pp(aa_nat_bool(P2,X3))
            & ! [Y4: nat] :
                ( pp(aa_nat_bool(P2,Y4))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y4),X3)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_236_nat__le__linear,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ).

% nat_le_linear
tff(fact_237_le__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( M2 = N ) ) ) ).

% le_antisym
tff(fact_238_eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = N )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% eq_imp_le
tff(fact_239_le0,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% le0
tff(fact_240_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_241_le__0__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_242_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_243_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_244_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% less_eq_nat.simps(1)
tff(fact_245_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M: nat] : N = aa_nat_nat(suc,M) ) ).

% not0_implies_Suc
tff(fact_246_Zero__not__Suc,axiom,
    ! [M2: nat] : zero_zero_nat != aa_nat_nat(suc,M2) ).

% Zero_not_Suc
tff(fact_247_Zero__neq__Suc,axiom,
    ! [M2: nat] : zero_zero_nat != aa_nat_nat(suc,M2) ).

% Zero_neq_Suc
tff(fact_248_Suc__neq__Zero,axiom,
    ! [M2: nat] : aa_nat_nat(suc,M2) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_249_zero__induct,axiom,
    ! [P2: fun_nat_bool,K2: nat] :
      ( pp(aa_nat_bool(P2,K2))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(P2,aa_nat_nat(suc,N3)))
           => pp(aa_nat_bool(P2,N3)) )
       => pp(aa_nat_bool(P2,zero_zero_nat)) ) ) ).

% zero_induct
tff(fact_250_diff__induct,axiom,
    ! [P2: fun_nat_fun_nat_bool,M2: nat,N: nat] :
      ( ! [X3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,X3),zero_zero_nat))
     => ( ! [Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,zero_zero_nat),aa_nat_nat(suc,Y3)))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,aa_nat_nat(suc,X3)),aa_nat_nat(suc,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,M2),N)) ) ) ) ).

% diff_induct
tff(fact_251_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N3: nat] : X != aa_nat_nat(suc,N3) ) ).

% list_decode.cases
tff(fact_252_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_253_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_254_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_255_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_256_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_257_nat__induct,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P2,zero_zero_nat))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(P2,N3))
           => pp(aa_nat_bool(P2,aa_nat_nat(suc,N3))) )
       => pp(aa_nat_bool(P2,N)) ) ) ).

% nat_induct
tff(fact_258_case4_I12_J,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(sa),deg)) ).

% case4(12)
tff(fact_259_le__zero__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_260_option_Osize_I4_J,axiom,
    ! [X2: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_261_option_Osize_I4_J,axiom,
    ! [X2: nat] : size_size_option_nat(some_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_262_option_Osize_I4_J,axiom,
    ! [X2: num] : size_size_option_num(some_num(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_263_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_264_exists__least__lemma,axiom,
    ! [P2: fun_nat_bool] :
      ( ~ pp(aa_nat_bool(P2,zero_zero_nat))
     => ( ? [X_1: nat] : pp(aa_nat_bool(P2,X_1))
       => ? [N3: nat] :
            ( ~ pp(aa_nat_bool(P2,N3))
            & pp(aa_nat_bool(P2,aa_nat_nat(suc,N3))) ) ) ) ).

% exists_least_lemma
tff(fact_265_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_266_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_267_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_268_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_269_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_270_divides__aux__eq,axiom,
    ! [Q2: code_integer,R: code_integer] :
      ( pp(aa_Pro5843519176478241656r_bool(unique5706413561485394159nteger,produc1086072967326762835nteger(Q2,R)))
    <=> ( R = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
tff(fact_271_divides__aux__eq,axiom,
    ! [Q2: nat,R: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R)))
    <=> ( R = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_272_divides__aux__eq,axiom,
    ! [Q2: int,R: int] :
      ( pp(aa_Pro4501881432599481050t_bool(unique6319869463603278526ux_int,product_Pair_int_int(Q2,R)))
    <=> ( R = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_273_one__le__mult__iff,axiom,
    ! [M2: nat,N: 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,M2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ) ) ).

% one_le_mult_iff
tff(fact_274__C1_C_I1_J,axiom,
    sa = vEBT_Leaf(a,b) ).

% "1"(1)
tff(fact_275_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_276_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).

% valid_0_not
tff(fact_277_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_278_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_279_insert_H__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_VEBT_insert(T,X)),N)) ) ).

% insert'_pres_valid
tff(fact_280_mult__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) = zero_zero_nat )
    <=> ( ( M2 = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_281_mult__0__right,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_282_mult__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N) )
    <=> ( ( M2 = N )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_283_mult__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),K2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K2) )
    <=> ( ( M2 = N )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_284_prod__decode__inverse,axiom,
    ! [N: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) = N ).

% prod_decode_inverse
tff(fact_285_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_286_one__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) )
    <=> ( ( M2 = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_287_mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M2 = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_288_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% mult.left_commute
tff(fact_289_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ).

% mult.left_commute
tff(fact_290_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: 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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% mult.left_commute
tff(fact_291_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C2: 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),C2)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% mult.left_commute
tff(fact_292_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_293_mult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A) ).

% mult.commute
tff(fact_294_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_295_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_296_mult_Oassoc,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% mult.assoc
tff(fact_297_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ).

% mult.assoc
tff(fact_298_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C2: 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)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% mult.assoc
tff(fact_299_mult_Oassoc,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% mult.assoc
tff(fact_300_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_301_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_302_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C2: 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)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_303_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_304_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_305_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_306_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,X) != aa_list_nat_nat(size_size_list_nat,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_307_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( size_size_list_int(X) != size_size_list_int(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_308_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_309_Suc__mult__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),N) )
    <=> ( M2 = N ) ) ).

% Suc_mult_cancel1
tff(fact_310_mult__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N) = zero_zero_nat ).

% mult_0
tff(fact_311_mult__le__mono2,axiom,
    ! [I2: nat,J2: nat,K2: 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_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),I2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),J2))) ) ).

% mult_le_mono2
tff(fact_312_mult__le__mono1,axiom,
    ! [I2: nat,J2: nat,K2: 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_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),K2))) ) ).

% mult_le_mono1
tff(fact_313_mult__le__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: 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_eq_nat,K2),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,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),L))) ) ) ).

% mult_le_mono
tff(fact_314_le__square,axiom,
    ! [M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),M2))) ).

% le_square
tff(fact_315_le__cube,axiom,
    ! [M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),M2)))) ).

% le_cube
tff(fact_316_Suc__mult__le__cancel1,axiom,
    ! [K2: nat,M2: nat,N: 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,K2)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_317_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
    <=> ( X = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_318_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
    <=> ( X = zero_zero_real ) ) ).

% zero_reorient
tff(fact_319_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
    <=> ( X = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_320_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_321_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_322_mult__cancel__right,axiom,
    ! [A: complex,C2: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C2) = aa_complex_complex(times_times_complex(B),C2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_323_mult__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_324_mult__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_325_mult__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2) )
    <=> ( ( C2 = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_326_mult__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C2) )
    <=> ( ( C2 = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_327_mult__cancel__left,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(C2),A) = aa_complex_complex(times_times_complex(C2),B) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_328_mult__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_329_mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_330_mult__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B) )
    <=> ( ( C2 = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_331_mult__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B) )
    <=> ( ( C2 = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_332_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_333_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_334_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_335_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_336_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_337_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_338_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_339_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_340_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_341_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_342_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_343_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_344_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_345_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_346_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_347_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_348_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_349_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C2: 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),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_350_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_351_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_352_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_353_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_354_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(aa_rat_fun_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_355_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_356_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_357_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(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_358_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_359_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_360_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_361_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(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_362_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_363_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_364_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_365_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(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_366_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_367_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_368_VEBT_Oinject_I2_J,axiom,
    ! [X21: bool,X22: bool,Y21: bool,Y22: bool] :
      ( ( vEBT_Leaf(X21,X22) = vEBT_Leaf(Y21,Y22) )
    <=> ( ( pp(X21)
        <=> pp(Y21) )
        & ( pp(X22)
        <=> pp(Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_369_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_370_VEBT_Osize_I4_J,axiom,
    ! [X21: bool,X22: bool] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf(X21,X22)) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_371_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_372_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_373_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_374_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_375_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_376_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_377_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_378_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_379_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_380_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_381_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_382_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_383_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_384_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_385_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_386_mult__left__cancel,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(C2),A) = aa_complex_complex(times_times_complex(C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_387_mult__left__cancel,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_388_mult__left__cancel,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_389_mult__left__cancel,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_390_mult__left__cancel,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_391_mult__right__cancel,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C2) = aa_complex_complex(times_times_complex(B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_392_mult__right__cancel,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_393_mult__right__cancel,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_394_mult__right__cancel,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_395_mult__right__cancel,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_396_mult__mono,axiom,
    ! [A: real,B: real,C2: 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,C2),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),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_397_mult__mono,axiom,
    ! [A: rat,B: rat,C2: 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,C2),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),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_398_mult__mono,axiom,
    ! [A: nat,B: nat,C2: 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,C2),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),C2))
           => 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_399_mult__mono,axiom,
    ! [A: int,B: int,C2: 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,C2),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),C2))
           => 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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_400_mult__mono_H,axiom,
    ! [A: real,B: real,C2: 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,C2),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),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_401_mult__mono_H,axiom,
    ! [A: rat,B: rat,C2: 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,C2),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),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_402_mult__mono_H,axiom,
    ! [A: nat,B: nat,C2: 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,C2),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),C2))
           => 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_403_mult__mono_H,axiom,
    ! [A: int,B: int,C2: 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,C2),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),C2))
           => 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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_404_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_405_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(aa_rat_fun_rat_rat(times_times_rat,A),A))) ).

% zero_le_square
tff(fact_406_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_407_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_408_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(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ).

% split_mult_pos_le
tff(fact_409_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_410_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C2: 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,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))) ) ) ).

% mult_left_mono_neg
tff(fact_411_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C2: 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,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))) ) ) ).

% mult_left_mono_neg
tff(fact_412_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C2: 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,C2),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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))) ) ) ).

% mult_left_mono_neg
tff(fact_413_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_414_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(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_415_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_416_mult__left__mono,axiom,
    ! [A: real,B: real,C2: 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),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))) ) ) ).

% mult_left_mono
tff(fact_417_mult__left__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))) ) ) ).

% mult_left_mono
tff(fact_418_mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))) ) ) ).

% mult_left_mono
tff(fact_419_mult__left__mono,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))) ) ) ).

% mult_left_mono
tff(fact_420_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C2: 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,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))) ) ) ).

% mult_right_mono_neg
tff(fact_421_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: 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,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))) ) ) ).

% mult_right_mono_neg
tff(fact_422_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C2: 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,C2),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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))) ) ) ).

% mult_right_mono_neg
tff(fact_423_mult__right__mono,axiom,
    ! [A: real,B: real,C2: 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),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))) ) ) ).

% mult_right_mono
tff(fact_424_mult__right__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))) ) ) ).

% mult_right_mono
tff(fact_425_mult__right__mono,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))) ) ) ).

% mult_right_mono
tff(fact_426_mult__right__mono,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => 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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))) ) ) ).

% mult_right_mono
tff(fact_427_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_428_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(aa_rat_fun_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_429_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_430_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_431_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(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ).

% split_mult_neg_le
tff(fact_432_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_433_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_434_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_435_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(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_436_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_437_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_438_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_439_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_440_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_441_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
    <=> ? [A5: bool,B4: bool] : T = vEBT_Leaf(A5,B4) ) ).

% deg1Leaf
tff(fact_442_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_443_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( N = one_one_nat )
       => ? [A3: bool,B3: bool] : T = vEBT_Leaf(A3,B3) ) ) ).

% deg_1_Leafy
tff(fact_444_nat__mult__eq__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N) )
    <=> ( ( K2 = zero_zero_nat )
        | ( M2 = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_445_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% deg_not_0
tff(fact_446_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: bool,X22: bool] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf(X21,X22)) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_447_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,N))))
     => ? [Info: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] : Tree = vEBT_Node(Info,aa_nat_nat(suc,aa_nat_nat(suc,N)),TreeList,S) ) ).

% deg_SUcn_Node
tff(fact_448_case4_I13_J,axiom,
    vEBT_VEBT_set_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList2,summary2)) = vEBT_VEBT_set_vebt(sa) ).

% case4(13)
tff(fact_449_deg__deg__n,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary)),N))
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_450_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_451_not__gr__zero,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_452_mult__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer),A) = A ).

% mult_1
tff(fact_453_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_454_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_455_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).

% mult_1
tff(fact_456_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_457_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_458_mult_Oright__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),one_one_Code_integer) = A ).

% mult.right_neutral
tff(fact_459_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_460_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_461_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_462_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_463_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_464_Suc__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_less_eq
tff(fact_465_Suc__mono,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N))) ) ).

% Suc_mono
tff(fact_466_lessI,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,N))) ).

% lessI
tff(fact_467_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_468_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% neq0_conv
tff(fact_469_less__nat__zero__code,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_nat_zero_code
tff(fact_470_nat__1__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) )
    <=> ( ( M2 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_471_nat__mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) = one_one_nat )
    <=> ( ( M2 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_472_case4_I3_J,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(summary2),m)) ).

% case4(3)
tff(fact_473_mult__cancel__left1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( ( C2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B) )
    <=> ( ( C2 = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_left1
tff(fact_474_mult__cancel__left1,axiom,
    ! [C2: complex,B: complex] :
      ( ( C2 = aa_complex_complex(times_times_complex(C2),B) )
    <=> ( ( C2 = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_475_mult__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( ( C2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) )
    <=> ( ( C2 = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_476_mult__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( ( C2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B) )
    <=> ( ( C2 = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_477_mult__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( ( C2 = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B) )
    <=> ( ( C2 = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_478_mult__cancel__left2,axiom,
    ! [C2: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A) = C2 )
    <=> ( ( C2 = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_left2
tff(fact_479_mult__cancel__left2,axiom,
    ! [C2: complex,A: complex] :
      ( ( aa_complex_complex(times_times_complex(C2),A) = C2 )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_480_mult__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A) = C2 )
    <=> ( ( C2 = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_481_mult__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A) = C2 )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_482_mult__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) = C2 )
    <=> ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_483_mult__cancel__right1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( ( C2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2) )
    <=> ( ( C2 = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_right1
tff(fact_484_mult__cancel__right1,axiom,
    ! [C2: complex,B: complex] :
      ( ( C2 = aa_complex_complex(times_times_complex(B),C2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_485_mult__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( ( C2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_486_mult__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( ( C2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_487_mult__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( ( C2 = aa_int_int(aa_int_fun_int_int(times_times_int,B),C2) )
    <=> ( ( C2 = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_488_mult__cancel__right2,axiom,
    ! [A: code_integer,C2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2) = C2 )
    <=> ( ( C2 = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_right2
tff(fact_489_mult__cancel__right2,axiom,
    ! [A: complex,C2: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C2) = C2 )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_490_mult__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = C2 )
    <=> ( ( C2 = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_491_mult__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = C2 )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_492_mult__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) = C2 )
    <=> ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_493_zero__less__Suc,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,N))) ).

% zero_less_Suc
tff(fact_494_less__Suc0,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( N = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_495_nat__mult__less__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_496_mult__less__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% mult_less_cancel2
tff(fact_497_nat__0__less__mult__iff,axiom,
    ! [M2: nat,N: 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,M2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_498_less__one,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),one_one_nat))
    <=> ( N = zero_zero_nat ) ) ).

% less_one
tff(fact_499_nat__mult__le__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: 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,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_500_mult__le__cancel2,axiom,
    ! [M2: nat,K2: nat,N: 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,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% mult_le_cancel2
tff(fact_501_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_502_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_503_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_504_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_505_lt__ex,axiom,
    ! [X: real] :
    ? [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X)) ).

% lt_ex
tff(fact_506_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_507_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_508_gt__ex,axiom,
    ! [X: real] :
    ? [X_12: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X_12)) ).

% gt_ex
tff(fact_509_gt__ex,axiom,
    ! [X: rat] :
    ? [X_12: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),X_12)) ).

% gt_ex
tff(fact_510_gt__ex,axiom,
    ! [X: nat] :
    ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X_12)) ).

% gt_ex
tff(fact_511_gt__ex,axiom,
    ! [X: int] :
    ? [X_12: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X_12)) ).

% gt_ex
tff(fact_512_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_513_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_514_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_515_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_516_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_517_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_518_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_519_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_520_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_521_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_522_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_523_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_524_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( A = B )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2)) ) ) ).

% ord_eq_less_trans
tff(fact_525_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C2)) ) ) ).

% ord_eq_less_trans
tff(fact_526_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( A = B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C2)) ) ) ).

% ord_eq_less_trans
tff(fact_527_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C2)) ) ) ).

% ord_eq_less_trans
tff(fact_528_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C2)) ) ) ).

% ord_eq_less_trans
tff(fact_529_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( B = C2 )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2)) ) ) ).

% ord_less_eq_trans
tff(fact_530_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( B = C2 )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C2)) ) ) ).

% ord_less_eq_trans
tff(fact_531_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( ( B = C2 )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C2)) ) ) ).

% ord_less_eq_trans
tff(fact_532_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( ( B = C2 )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C2)) ) ) ).

% ord_less_eq_trans
tff(fact_533_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( ( B = C2 )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C2)) ) ) ).

% ord_less_eq_trans
tff(fact_534_less__induct,axiom,
    ! [P2: fun_nat_bool,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),X3))
             => pp(aa_nat_bool(P2,Y4)) )
         => pp(aa_nat_bool(P2,X3)) )
     => pp(aa_nat_bool(P2,A)) ) ).

% less_induct
tff(fact_535_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_536_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_537_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_538_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_539_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_540_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_541_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_542_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_543_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_544_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_545_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_546_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_547_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_548_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_549_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_550_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).

% dual_order.irrefl
tff(fact_551_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_552_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_553_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_554_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_555_exists__least__iff,axiom,
    ! [P2: fun_nat_bool] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P2,X_13))
    <=> ? [N4: nat] :
          ( pp(aa_nat_bool(P2,N4))
          & ! [M6: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M6),N4))
             => ~ pp(aa_nat_bool(P2,M6)) ) ) ) ).

% exists_least_iff
tff(fact_556_linorder__less__wlog,axiom,
    ! [P2: 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(P2,A3),B3)) )
     => ( ! [A3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(P2,A3),A3))
       => ( ! [A3: real,B3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P2,B3),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P2,A3),B3)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P2,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_557_linorder__less__wlog,axiom,
    ! [P2: 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(P2,A3),B3)) )
     => ( ! [A3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,A3),A3))
       => ( ! [A3: rat,B3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,B3),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,A3),B3)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_558_linorder__less__wlog,axiom,
    ! [P2: 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(P2,A3),B3)) )
     => ( ! [A3: num] : pp(aa_num_bool(aa_num_fun_num_bool(P2,A3),A3))
       => ( ! [A3: num,B3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(P2,B3),A3))
             => pp(aa_num_bool(aa_num_fun_num_bool(P2,A3),B3)) )
         => pp(aa_num_bool(aa_num_fun_num_bool(P2,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_559_linorder__less__wlog,axiom,
    ! [P2: 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(P2,A3),B3)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A3),A3))
       => ( ! [A3: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,B3),A3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A3),B3)) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_560_linorder__less__wlog,axiom,
    ! [P2: 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(P2,A3),B3)) )
     => ( ! [A3: int] : pp(aa_int_bool(aa_int_fun_int_bool(P2,A3),A3))
       => ( ! [A3: int,B3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(P2,B3),A3))
             => pp(aa_int_bool(aa_int_fun_int_bool(P2,A3),B3)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(P2,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_561_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2)) ) ) ).

% order.strict_trans
tff(fact_562_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C2)) ) ) ).

% order.strict_trans
tff(fact_563_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C2: 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),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C2)) ) ) ).

% order.strict_trans
tff(fact_564_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C2)) ) ) ).

% order.strict_trans
tff(fact_565_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C2)) ) ) ).

% order.strict_trans
tff(fact_566_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_567_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_568_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_569_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_570_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_571_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),A)) ) ) ).

% dual_order.strict_trans
tff(fact_572_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C2: 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,C2),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),A)) ) ) ).

% dual_order.strict_trans
tff(fact_573_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C2: 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,C2),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),A)) ) ) ).

% dual_order.strict_trans
tff(fact_574_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C2: 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,C2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),A)) ) ) ).

% dual_order.strict_trans
tff(fact_575_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C2: 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,C2),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),A)) ) ) ).

% dual_order.strict_trans
tff(fact_576_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_577_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_578_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_579_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_580_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_581_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_582_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_583_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_584_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_585_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_586_nat__neq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 != N )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% nat_neq_iff
tff(fact_587_less__not__refl,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_not_refl
tff(fact_588_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
     => ( M2 != N ) ) ).

% less_not_refl2
tff(fact_589_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S2),T))
     => ( S2 != T ) ) ).

% less_not_refl3
tff(fact_590_less__irrefl__nat,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_irrefl_nat
tff(fact_591_nat__less__induct,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N3))
             => pp(aa_nat_bool(P2,M3)) )
         => pp(aa_nat_bool(P2,N3)) )
     => pp(aa_nat_bool(P2,N)) ) ).

% nat_less_induct
tff(fact_592_infinite__descent,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( ! [N3: nat] :
          ( ~ pp(aa_nat_bool(P2,N3))
         => ? [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N3))
              & ~ pp(aa_nat_bool(P2,M3)) ) )
     => pp(aa_nat_bool(P2,N)) ) ).

% infinite_descent
tff(fact_593_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_594_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_595_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_596_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_597_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_598_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_599_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_600_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_601_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_602_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_603_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_604_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_605_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_606_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_607_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_608_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_609_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_610_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_611_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_612_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_613_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_614_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_615_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_616_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_617_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_618_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_619_ord__eq__less__subst,axiom,
    ! [A: real,F2: fun_real_real,B: real,C2: real] :
      ( ( A = aa_real_real(F2,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F2,X3)),aa_real_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F2,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_620_ord__eq__less__subst,axiom,
    ! [A: rat,F2: fun_real_rat,B: real,C2: real] :
      ( ( A = aa_real_rat(F2,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F2,X3)),aa_real_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F2,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_621_ord__eq__less__subst,axiom,
    ! [A: num,F2: fun_real_num,B: real,C2: real] :
      ( ( A = aa_real_num(F2,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F2,X3)),aa_real_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_real_num(F2,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_622_ord__eq__less__subst,axiom,
    ! [A: nat,F2: fun_real_nat,B: real,C2: real] :
      ( ( A = aa_real_nat(F2,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F2,X3)),aa_real_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_real_nat(F2,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_623_ord__eq__less__subst,axiom,
    ! [A: int,F2: fun_real_int,B: real,C2: real] :
      ( ( A = aa_real_int(F2,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F2,X3)),aa_real_int(F2,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_real_int(F2,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_624_ord__eq__less__subst,axiom,
    ! [A: real,F2: fun_rat_real,B: rat,C2: rat] :
      ( ( A = aa_rat_real(F2,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F2,X3)),aa_rat_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F2,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_625_ord__eq__less__subst,axiom,
    ! [A: rat,F2: fun_rat_rat,B: rat,C2: rat] :
      ( ( A = aa_rat_rat(F2,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F2,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_626_ord__eq__less__subst,axiom,
    ! [A: num,F2: fun_rat_num,B: rat,C2: rat] :
      ( ( A = aa_rat_num(F2,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F2,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_627_ord__eq__less__subst,axiom,
    ! [A: nat,F2: fun_rat_nat,B: rat,C2: rat] :
      ( ( A = aa_rat_nat(F2,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F2,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_628_ord__eq__less__subst,axiom,
    ! [A: int,F2: fun_rat_int,B: rat,C2: rat] :
      ( ( A = aa_rat_int(F2,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F2,X3)),aa_rat_int(F2,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F2,C2))) ) ) ) ).

% ord_eq_less_subst
tff(fact_629_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F2: fun_real_real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_real(F2,B) = C2 )
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F2,X3)),aa_real_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F2,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_630_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F2: fun_real_rat,C2: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_rat(F2,B) = C2 )
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F2,X3)),aa_real_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F2,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_631_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F2: fun_real_num,C2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_num(F2,B) = C2 )
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F2,X3)),aa_real_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F2,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_632_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F2: fun_real_nat,C2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_nat(F2,B) = C2 )
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F2,X3)),aa_real_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F2,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_633_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F2: fun_real_int,C2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_int(F2,B) = C2 )
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F2,X3)),aa_real_int(F2,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F2,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_634_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F2: fun_rat_real,C2: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_real(F2,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F2,X3)),aa_rat_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F2,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_635_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F2: fun_rat_rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_rat(F2,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F2,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_636_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F2: fun_rat_num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_num(F2,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F2,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_637_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F2: fun_rat_nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_nat(F2,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F2,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_638_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F2: fun_rat_int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_int(F2,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F2,X3)),aa_rat_int(F2,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F2,A)),C2)) ) ) ) ).

% ord_less_eq_subst
tff(fact_639_order__less__irrefl,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X)) ).

% order_less_irrefl
tff(fact_640_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_641_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_642_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_643_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_644_order__less__subst1,axiom,
    ! [A: real,F2: fun_real_real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F2,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F2,X3)),aa_real_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F2,C2))) ) ) ) ).

% order_less_subst1
tff(fact_645_order__less__subst1,axiom,
    ! [A: real,F2: fun_rat_real,B: rat,C2: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F2,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F2,X3)),aa_rat_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F2,C2))) ) ) ) ).

% order_less_subst1
tff(fact_646_order__less__subst1,axiom,
    ! [A: real,F2: fun_num_real,B: num,C2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F2,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F2,X3)),aa_num_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F2,C2))) ) ) ) ).

% order_less_subst1
tff(fact_647_order__less__subst1,axiom,
    ! [A: real,F2: fun_nat_real,B: nat,C2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F2,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C2))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F2,X3)),aa_nat_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F2,C2))) ) ) ) ).

% order_less_subst1
tff(fact_648_order__less__subst1,axiom,
    ! [A: real,F2: fun_int_real,B: int,C2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F2,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C2))
       => ( ! [X3: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F2,X3)),aa_int_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F2,C2))) ) ) ) ).

% order_less_subst1
tff(fact_649_order__less__subst1,axiom,
    ! [A: rat,F2: fun_real_rat,B: real,C2: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F2,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F2,X3)),aa_real_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F2,C2))) ) ) ) ).

% order_less_subst1
tff(fact_650_order__less__subst1,axiom,
    ! [A: rat,F2: fun_rat_rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F2,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F2,C2))) ) ) ) ).

% order_less_subst1
tff(fact_651_order__less__subst1,axiom,
    ! [A: rat,F2: fun_num_rat,B: num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F2,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F2,C2))) ) ) ) ).

% order_less_subst1
tff(fact_652_order__less__subst1,axiom,
    ! [A: rat,F2: fun_nat_rat,B: nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F2,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C2))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F2,X3)),aa_nat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F2,C2))) ) ) ) ).

% order_less_subst1
tff(fact_653_order__less__subst1,axiom,
    ! [A: rat,F2: fun_int_rat,B: int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F2,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C2))
       => ( ! [X3: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F2,X3)),aa_int_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F2,C2))) ) ) ) ).

% order_less_subst1
tff(fact_654_order__less__subst2,axiom,
    ! [A: real,B: real,F2: fun_real_real,C2: 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(F2,B)),C2))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F2,X3)),aa_real_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F2,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_655_order__less__subst2,axiom,
    ! [A: real,B: real,F2: fun_real_rat,C2: 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(F2,B)),C2))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F2,X3)),aa_real_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F2,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_656_order__less__subst2,axiom,
    ! [A: real,B: real,F2: fun_real_num,C2: 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(F2,B)),C2))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F2,X3)),aa_real_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F2,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_657_order__less__subst2,axiom,
    ! [A: real,B: real,F2: fun_real_nat,C2: 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(F2,B)),C2))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F2,X3)),aa_real_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F2,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_658_order__less__subst2,axiom,
    ! [A: real,B: real,F2: fun_real_int,C2: 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(F2,B)),C2))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F2,X3)),aa_real_int(F2,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F2,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_659_order__less__subst2,axiom,
    ! [A: rat,B: rat,F2: fun_rat_real,C2: 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(F2,B)),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F2,X3)),aa_rat_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F2,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_660_order__less__subst2,axiom,
    ! [A: rat,B: rat,F2: fun_rat_rat,C2: 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(F2,B)),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F2,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_661_order__less__subst2,axiom,
    ! [A: rat,B: rat,F2: fun_rat_num,C2: 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(F2,B)),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F2,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_662_order__less__subst2,axiom,
    ! [A: rat,B: rat,F2: fun_rat_nat,C2: 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(F2,B)),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F2,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_663_order__less__subst2,axiom,
    ! [A: rat,B: rat,F2: fun_rat_int,C2: 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(F2,B)),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F2,X3)),aa_rat_int(F2,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F2,A)),C2)) ) ) ) ).

% order_less_subst2
tff(fact_664_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_665_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_666_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_667_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_668_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_669_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P2: 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(P2) ) ) ).

% order_less_imp_triv
tff(fact_670_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P2: 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(P2) ) ) ).

% order_less_imp_triv
tff(fact_671_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P2: 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(P2) ) ) ).

% order_less_imp_triv
tff(fact_672_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P2: 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(P2) ) ) ).

% order_less_imp_triv
tff(fact_673_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P2: 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(P2) ) ) ).

% order_less_imp_triv
tff(fact_674_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_675_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_676_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_677_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_678_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_679_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_680_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_681_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_682_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_683_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_684_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_685_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_686_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_687_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_688_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_689_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_690_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_691_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_692_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_693_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_694_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),one_one_Code_integer)) ).

% less_numeral_extra(4)
tff(fact_695_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_696_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_697_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_698_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_699_one__reorient,axiom,
    ! [X: code_integer] :
      ( ( one_one_Code_integer = X )
    <=> ( X = one_one_Code_integer ) ) ).

% one_reorient
tff(fact_700_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
    <=> ( X = one_one_complex ) ) ).

% one_reorient
tff(fact_701_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
    <=> ( X = one_one_real ) ) ).

% one_reorient
tff(fact_702_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_703_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_704_less__numeral__extra_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer)) ).

% less_numeral_extra(1)
tff(fact_705_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_706_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_707_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_708_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_709_not__one__less__zero,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),zero_z3403309356797280102nteger)) ).

% not_one_less_zero
tff(fact_710_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_711_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_712_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_713_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_714_zero__less__one,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer)) ).

% zero_less_one
tff(fact_715_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_716_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_717_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_718_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_719_lift__Suc__mono__less__iff,axiom,
    ! [F2: fun_nat_real,N: nat,M2: nat] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F2,N3)),aa_nat_real(F2,aa_nat_nat(suc,N3))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F2,N)),aa_nat_real(F2,M2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_720_lift__Suc__mono__less__iff,axiom,
    ! [F2: fun_nat_rat,N: nat,M2: nat] :
      ( ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F2,N3)),aa_nat_rat(F2,aa_nat_nat(suc,N3))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F2,N)),aa_nat_rat(F2,M2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_721_lift__Suc__mono__less__iff,axiom,
    ! [F2: fun_nat_num,N: nat,M2: nat] :
      ( ! [N3: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F2,N3)),aa_nat_num(F2,aa_nat_nat(suc,N3))))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F2,N)),aa_nat_num(F2,M2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_722_lift__Suc__mono__less__iff,axiom,
    ! [F2: fun_nat_nat,N: nat,M2: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F2,N3)),aa_nat_nat(F2,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F2,N)),aa_nat_nat(F2,M2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_723_lift__Suc__mono__less__iff,axiom,
    ! [F2: fun_nat_int,N: nat,M2: nat] :
      ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F2,N3)),aa_nat_int(F2,aa_nat_nat(suc,N3))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F2,N)),aa_nat_int(F2,M2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_724_lift__Suc__mono__less,axiom,
    ! [F2: fun_nat_real,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F2,N3)),aa_nat_real(F2,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F2,N)),aa_nat_real(F2,N2))) ) ) ).

% lift_Suc_mono_less
tff(fact_725_lift__Suc__mono__less,axiom,
    ! [F2: fun_nat_rat,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F2,N3)),aa_nat_rat(F2,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F2,N)),aa_nat_rat(F2,N2))) ) ) ).

% lift_Suc_mono_less
tff(fact_726_lift__Suc__mono__less,axiom,
    ! [F2: fun_nat_num,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F2,N3)),aa_nat_num(F2,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F2,N)),aa_nat_num(F2,N2))) ) ) ).

% lift_Suc_mono_less
tff(fact_727_lift__Suc__mono__less,axiom,
    ! [F2: fun_nat_nat,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F2,N3)),aa_nat_nat(F2,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F2,N)),aa_nat_nat(F2,N2))) ) ) ).

% lift_Suc_mono_less
tff(fact_728_lift__Suc__mono__less,axiom,
    ! [F2: fun_nat_int,N: nat,N2: nat] :
      ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F2,N3)),aa_nat_int(F2,aa_nat_nat(suc,N3))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F2,N)),aa_nat_int(F2,N2))) ) ) ).

% lift_Suc_mono_less
tff(fact_729_less__1__mult,axiom,
    ! [M2: code_integer,N: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),M2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),N))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,M2),N))) ) ) ).

% less_1_mult
tff(fact_730_less__1__mult,axiom,
    ! [M2: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),M2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M2),N))) ) ) ).

% less_1_mult
tff(fact_731_less__1__mult,axiom,
    ! [M2: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),M2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M2),N))) ) ) ).

% less_1_mult
tff(fact_732_less__1__mult,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
       => 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,M2),N))) ) ) ).

% less_1_mult
tff(fact_733_less__1__mult,axiom,
    ! [M2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),M2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),N))
       => 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,M2),N))) ) ) ).

% less_1_mult
tff(fact_734_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_735_nat__induct__non__zero,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(P2,one_one_nat))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
             => ( pp(aa_nat_bool(P2,N3))
               => pp(aa_nat_bool(P2,aa_nat_nat(suc,N3))) ) )
         => pp(aa_nat_bool(P2,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_736_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_737_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_738_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_739_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_740_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_741_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_742_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_743_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_744_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_745_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_746_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_747_order__less__le__subst2,axiom,
    ! [A: real,B: real,F2: fun_real_real,C2: 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(F2,B)),C2))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F2,X3)),aa_real_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F2,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_748_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F2: fun_rat_real,C2: 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(F2,B)),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F2,X3)),aa_rat_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F2,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_749_order__less__le__subst2,axiom,
    ! [A: num,B: num,F2: fun_num_real,C2: 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(F2,B)),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F2,X3)),aa_num_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F2,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_750_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F2: fun_nat_real,C2: 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(F2,B)),C2))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F2,X3)),aa_nat_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F2,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_751_order__less__le__subst2,axiom,
    ! [A: int,B: int,F2: fun_int_real,C2: 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(F2,B)),C2))
       => ( ! [X3: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F2,X3)),aa_int_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F2,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_752_order__less__le__subst2,axiom,
    ! [A: real,B: real,F2: fun_real_rat,C2: 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(F2,B)),C2))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F2,X3)),aa_real_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F2,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_753_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F2: fun_rat_rat,C2: 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(F2,B)),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F2,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_754_order__less__le__subst2,axiom,
    ! [A: num,B: num,F2: fun_num_rat,C2: 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(F2,B)),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F2,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_755_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F2: fun_nat_rat,C2: 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(F2,B)),C2))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F2,X3)),aa_nat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F2,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_756_order__less__le__subst2,axiom,
    ! [A: int,B: int,F2: fun_int_rat,C2: 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(F2,B)),C2))
       => ( ! [X3: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F2,X3)),aa_int_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F2,A)),C2)) ) ) ) ).

% order_less_le_subst2
tff(fact_757_order__less__le__subst1,axiom,
    ! [A: real,F2: fun_rat_real,B: rat,C2: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F2,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F2,X3)),aa_rat_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F2,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_758_order__less__le__subst1,axiom,
    ! [A: rat,F2: fun_rat_rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F2,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F2,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_759_order__less__le__subst1,axiom,
    ! [A: num,F2: fun_rat_num,B: rat,C2: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F2,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F2,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_760_order__less__le__subst1,axiom,
    ! [A: nat,F2: fun_rat_nat,B: rat,C2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F2,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F2,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_761_order__less__le__subst1,axiom,
    ! [A: int,F2: fun_rat_int,B: rat,C2: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F2,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F2,X3)),aa_rat_int(F2,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F2,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_762_order__less__le__subst1,axiom,
    ! [A: real,F2: fun_num_real,B: num,C2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F2,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F2,X3)),aa_num_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F2,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_763_order__less__le__subst1,axiom,
    ! [A: rat,F2: fun_num_rat,B: num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F2,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F2,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_764_order__less__le__subst1,axiom,
    ! [A: num,F2: fun_num_num,B: num,C2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F2,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F2,X3)),aa_num_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F2,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_765_order__less__le__subst1,axiom,
    ! [A: nat,F2: fun_num_nat,B: num,C2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F2,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F2,X3)),aa_num_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F2,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_766_order__less__le__subst1,axiom,
    ! [A: int,F2: fun_num_int,B: num,C2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F2,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F2,X3)),aa_num_int(F2,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F2,C2))) ) ) ) ).

% order_less_le_subst1
tff(fact_767_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F2: fun_rat_real,C2: 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(F2,B)),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F2,X3)),aa_rat_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F2,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_768_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F2: fun_rat_rat,C2: 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(F2,B)),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F2,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_769_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F2: fun_rat_num,C2: 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(F2,B)),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F2,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_770_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F2: fun_rat_nat,C2: 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(F2,B)),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F2,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_771_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F2: fun_rat_int,C2: 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(F2,B)),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F2,X3)),aa_rat_int(F2,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F2,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_772_order__le__less__subst2,axiom,
    ! [A: num,B: num,F2: fun_num_real,C2: 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(F2,B)),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F2,X3)),aa_num_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F2,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_773_order__le__less__subst2,axiom,
    ! [A: num,B: num,F2: fun_num_rat,C2: 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(F2,B)),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F2,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_774_order__le__less__subst2,axiom,
    ! [A: num,B: num,F2: fun_num_num,C2: 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(F2,B)),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F2,X3)),aa_num_num(F2,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F2,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_775_order__le__less__subst2,axiom,
    ! [A: num,B: num,F2: fun_num_nat,C2: 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(F2,B)),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F2,X3)),aa_num_nat(F2,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F2,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_776_order__le__less__subst2,axiom,
    ! [A: num,B: num,F2: fun_num_int,C2: 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(F2,B)),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F2,X3)),aa_num_int(F2,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F2,A)),C2)) ) ) ) ).

% order_le_less_subst2
tff(fact_777_order__le__less__subst1,axiom,
    ! [A: real,F2: fun_real_real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(F2,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F2,X3)),aa_real_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F2,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_778_order__le__less__subst1,axiom,
    ! [A: real,F2: fun_rat_real,B: rat,C2: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_rat_real(F2,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F2,X3)),aa_rat_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F2,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_779_order__le__less__subst1,axiom,
    ! [A: real,F2: fun_num_real,B: num,C2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_num_real(F2,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F2,X3)),aa_num_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F2,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_780_order__le__less__subst1,axiom,
    ! [A: real,F2: fun_nat_real,B: nat,C2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(F2,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C2))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F2,X3)),aa_nat_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F2,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_781_order__le__less__subst1,axiom,
    ! [A: real,F2: fun_int_real,B: int,C2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_int_real(F2,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C2))
       => ( ! [X3: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F2,X3)),aa_int_real(F2,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F2,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_782_order__le__less__subst1,axiom,
    ! [A: rat,F2: fun_real_rat,B: real,C2: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_real_rat(F2,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C2))
       => ( ! [X3: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F2,X3)),aa_real_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F2,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_783_order__le__less__subst1,axiom,
    ! [A: rat,F2: fun_rat_rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F2,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C2))
       => ( ! [X3: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F2,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_784_order__le__less__subst1,axiom,
    ! [A: rat,F2: fun_num_rat,B: num,C2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F2,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C2))
       => ( ! [X3: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F2,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_785_order__le__less__subst1,axiom,
    ! [A: rat,F2: fun_nat_rat,B: nat,C2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F2,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C2))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F2,X3)),aa_nat_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F2,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_786_order__le__less__subst1,axiom,
    ! [A: rat,F2: fun_int_rat,B: int,C2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F2,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C2))
       => ( ! [X3: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F2,X3)),aa_int_rat(F2,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F2,C2))) ) ) ) ).

% order_le_less_subst1
tff(fact_787_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_788_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_789_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_790_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_791_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_792_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_793_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_794_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_795_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_796_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_797_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_798_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_799_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_800_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_801_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_802_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_803_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_804_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_805_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_806_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_807_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_808_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_809_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_810_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_811_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_812_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_813_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_814_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_815_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_816_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_817_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_818_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_819_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_820_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_821_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_822_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_823_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_824_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_825_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_826_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_827_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_828_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_829_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_830_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_831_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_832_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_833_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_834_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_835_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_836_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_837_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_838_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_839_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_840_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_841_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_842_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_843_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_844_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_845_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_846_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_847_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_848_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_849_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_850_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_851_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_852_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_853_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_854_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_855_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_856_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_857_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_858_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C2),B))
       => pp(aa_set_nat_bool(ord_less_set_nat(C2),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_859_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C2: 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,C2),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_860_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C2: 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,C2),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_861_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: 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,C2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_862_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C2: 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,C2),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_863_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_864_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( pp(aa_set_nat_bool(ord_less_set_nat(C2),B))
       => pp(aa_set_nat_bool(ord_less_set_nat(C2),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_865_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C2: 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,C2),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_866_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C2: 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,C2),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C2),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_867_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: 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,C2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_868_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C2: 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,C2),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_869_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_870_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_871_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_872_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_873_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_874_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_875_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_876_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_877_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_878_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_879_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_880_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_881_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_882_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_883_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_884_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_885_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_886_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_887_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_888_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_889_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_890_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_891_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C2: 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),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2)) ) ) ).

% order.strict_trans2
tff(fact_892_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),C2))
       => pp(aa_set_nat_bool(ord_less_set_nat(A),C2)) ) ) ).

% order.strict_trans2
tff(fact_893_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C2)) ) ) ).

% order.strict_trans2
tff(fact_894_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C2: 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),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C2)) ) ) ).

% order.strict_trans2
tff(fact_895_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C2)) ) ) ).

% order.strict_trans2
tff(fact_896_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C2)) ) ) ).

% order.strict_trans2
tff(fact_897_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C2: 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),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2)) ) ) ).

% order.strict_trans1
tff(fact_898_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( pp(aa_set_nat_bool(ord_less_set_nat(B),C2))
       => pp(aa_set_nat_bool(ord_less_set_nat(A),C2)) ) ) ).

% order.strict_trans1
tff(fact_899_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C2)) ) ) ).

% order.strict_trans1
tff(fact_900_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C2: 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),C2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C2)) ) ) ).

% order.strict_trans1
tff(fact_901_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C2)) ) ) ).

% order.strict_trans1
tff(fact_902_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C2)) ) ) ).

% order.strict_trans1
tff(fact_903_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_904_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_905_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_906_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_907_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_908_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_909_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_910_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_911_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_912_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_913_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_914_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_915_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_916_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_917_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_918_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_919_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_920_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_921_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_922_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_923_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_924_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_925_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_926_dense__le,axiom,
    ! [Y: real,Z2: real] :
      ( ! [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Z2)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z2)) ) ).

% dense_le
tff(fact_927_dense__le,axiom,
    ! [Y: rat,Z2: rat] :
      ( ! [X3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Z2)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z2)) ) ).

% dense_le
tff(fact_928_dense__ge,axiom,
    ! [Z2: real,Y: real] :
      ( ! [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X3)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z2)) ) ).

% dense_ge
tff(fact_929_dense__ge,axiom,
    ! [Z2: rat,Y: rat] :
      ( ! [X3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z2)) ) ).

% dense_ge
tff(fact_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_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_938_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_939_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_940_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_941_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_942_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_943_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_944_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_945_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_946_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_947_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_948_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_949_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_950_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_951_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_952_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_953_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_954_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_955_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_956_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_957_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_958_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_959_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_960_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_961_not__less__zero,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less_zero
tff(fact_962_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr_zeroI
tff(fact_963_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_964_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_965_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_966_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_967_not__less__less__Suc__eq,axiom,
    ! [N: nat,M2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M2)))
      <=> ( N = M2 ) ) ) ).

% not_less_less_Suc_eq
tff(fact_968_strict__inc__induct,axiom,
    ! [I2: nat,J2: nat,P2: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
     => ( ! [I3: nat] :
            ( ( J2 = aa_nat_nat(suc,I3) )
           => pp(aa_nat_bool(P2,I3)) )
       => ( ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J2))
             => ( pp(aa_nat_bool(P2,aa_nat_nat(suc,I3)))
               => pp(aa_nat_bool(P2,I3)) ) )
         => pp(aa_nat_bool(P2,I2)) ) ) ) ).

% strict_inc_induct
tff(fact_969_less__Suc__induct,axiom,
    ! [I2: nat,J2: nat,P2: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
     => ( ! [I3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,I3),aa_nat_nat(suc,I3)))
       => ( ! [I3: nat,J3: nat,K: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),K))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,I3),J3))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,J3),K))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,I3),K)) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,I2),J2)) ) ) ) ).

% less_Suc_induct
tff(fact_970_less__trans__Suc,axiom,
    ! [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(ord_less_nat,aa_nat_nat(suc,I2)),K2)) ) ) ).

% less_trans_Suc
tff(fact_971_Suc__less__SucD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_less_SucD
tff(fact_972_less__antisym,axiom,
    ! [N: nat,M2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M2)))
       => ( M2 = N ) ) ) ).

% less_antisym
tff(fact_973_Suc__less__eq2,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
    <=> ? [M7: nat] :
          ( ( M2 = aa_nat_nat(suc,M7) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M7)) ) ) ).

% Suc_less_eq2
tff(fact_974_All__less__Suc,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P2,I)) )
    <=> ( pp(aa_nat_bool(P2,N))
        & ! [I: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
           => pp(aa_nat_bool(P2,I)) ) ) ) ).

% All_less_Suc
tff(fact_975_not__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M2))) ) ).

% not_less_eq
tff(fact_976_less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | ( M2 = N ) ) ) ).

% less_Suc_eq
tff(fact_977_Ex__less__Suc,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P2,I)) )
    <=> ( pp(aa_nat_bool(P2,N))
        | ? [I: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
            & pp(aa_nat_bool(P2,I)) ) ) ) ).

% Ex_less_Suc
tff(fact_978_less__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N))) ) ).

% less_SucI
tff(fact_979_less__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => ( M2 = N ) ) ) ).

% less_SucE
tff(fact_980_Suc__lessI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( ( aa_nat_nat(suc,M2) != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M2)),N)) ) ) ).

% Suc_lessI
tff(fact_981_Suc__lessE,axiom,
    ! [I2: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I2)),K2))
     => ~ ! [J3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J3))
           => ( K2 != aa_nat_nat(suc,J3) ) ) ) ).

% Suc_lessE
tff(fact_982_Suc__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M2)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_lessD
tff(fact_983_Nat_OlessE,axiom,
    ! [I2: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K2))
     => ( ( K2 != aa_nat_nat(suc,I2) )
       => ~ ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J3))
             => ( K2 != aa_nat_nat(suc,J3) ) ) ) ) ).

% Nat.lessE
tff(fact_984_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: bool,X22: bool] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf(X21,X22) ).

% VEBT.distinct(1)
tff(fact_985_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,X222: bool] : Y != vEBT_Leaf(X212,X222) ) ).

% VEBT.exhaust
tff(fact_986_le__numeral__extra_I4_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),one_one_Code_integer)) ).

% le_numeral_extra(4)
tff(fact_987_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_988_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_989_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_990_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_991_zero__neq__one,axiom,
    zero_z3403309356797280102nteger != one_one_Code_integer ).

% zero_neq_one
tff(fact_992_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_993_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_994_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_995_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_996_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_997_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_998_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr0I
tff(fact_999_not__gr0,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr0
tff(fact_1000_not__less0,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less0
tff(fact_1001_less__zeroE,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_zeroE
tff(fact_1002_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_1003_infinite__descent0,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P2,zero_zero_nat))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
           => ( ~ pp(aa_nat_bool(P2,N3))
             => ? [M3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N3))
                  & ~ pp(aa_nat_bool(P2,M3)) ) ) )
       => pp(aa_nat_bool(P2,N)) ) ) ).

% infinite_descent0
tff(fact_1004_mult_Ocomm__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),one_one_Code_integer) = A ).

% mult.comm_neutral
tff(fact_1005_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_1006_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_1007_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_1008_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_1009_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_1010_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_1011_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_1012_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_1013_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_1014_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_1015_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_1016_less__mono__imp__le__mono,axiom,
    ! [F2: fun_nat_nat,I2: nat,J2: nat] :
      ( ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F2,I3)),aa_nat_nat(F2,J3))) )
     => ( 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_eq_nat,aa_nat_nat(F2,I2)),aa_nat_nat(F2,J2))) ) ) ).

% less_mono_imp_le_mono
tff(fact_1017_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( ( M2 != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% le_neq_implies_less
tff(fact_1018_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | ( M2 = N ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% less_or_eq_imp_le
tff(fact_1019_le__eq__less__or__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | ( M2 = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_1020_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% less_imp_le_nat
tff(fact_1021_nat__less__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
        & ( M2 != N ) ) ) ).

% nat_less_le
tff(fact_1022_nat__mult__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ).

% nat_mult_1
tff(fact_1023_nat__mult__1__right,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ).

% nat_mult_1_right
tff(fact_1024_nat__mult__eq__cancel1,axiom,
    ! [K2: nat,M2: nat,N: 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(times_times_nat,K2),M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N) )
      <=> ( M2 = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1025_nat__mult__less__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_1026_mult__less__cancel__right2,axiom,
    ! [A: code_integer,C2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),C2))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),one_one_Code_integer)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C2),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1027_mult__less__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),C2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => 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,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1028_mult__less__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),C2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => 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,C2),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_1029_mult__less__cancel__right2,axiom,
    ! [A: int,C2: 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),C2)),C2))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => 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,C2),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_1030_mult__less__cancel__right1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C2),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),one_one_Code_integer)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1031_mult__less__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => 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,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1032_mult__less__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => 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,C2),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_1033_mult__less__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => 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,C2),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_1034_mult__less__cancel__left2,axiom,
    ! [C2: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),C2))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),one_one_Code_integer)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C2),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1035_mult__less__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),C2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => 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,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1036_mult__less__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),C2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => 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,C2),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_1037_mult__less__cancel__left2,axiom,
    ! [C2: 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,C2),A)),C2))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => 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,C2),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_1038_mult__less__cancel__left1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C2),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),one_one_Code_integer)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1039_mult__less__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => 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,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1040_mult__less__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => 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,C2),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_1041_mult__less__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => 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,C2),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_1042_mult__le__cancel__right2,axiom,
    ! [A: code_integer,C2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),C2))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C2),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1043_mult__le__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),C2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => 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,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1044_mult__le__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),C2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => 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,C2),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_1045_mult__le__cancel__right2,axiom,
    ! [A: int,C2: 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),C2)),C2))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => 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,C2),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_1046_mult__le__cancel__right1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),B)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C2),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),one_one_Code_integer)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1047_mult__le__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => 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,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1048_mult__le__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => 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,C2),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_1049_mult__le__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => 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,C2),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_1050_mult__le__cancel__left2,axiom,
    ! [C2: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),C2))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C2),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1051_mult__le__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),C2))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => 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,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1052_mult__le__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),C2))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => 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,C2),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_1053_mult__le__cancel__left2,axiom,
    ! [C2: 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,C2),A)),C2))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => 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,C2),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_1054_mult__le__cancel__left1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),B)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C2),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),one_one_Code_integer)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1055_mult__le__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => 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,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1056_mult__le__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => 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,C2),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_1057_mult__le__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => 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,C2),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_1058_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_1059_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(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% mult_neg_neg
tff(fact_1060_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_1061_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_1062_not__square__less__zero,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),zero_zero_rat)) ).

% not_square_less_zero
tff(fact_1063_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_1064_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_1065_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_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_1066_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_1067_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_1068_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(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ) ).

% mult_neg_pos
tff(fact_1069_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_1070_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_1071_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_1072_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(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ) ).

% mult_pos_neg
tff(fact_1073_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_1074_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_1075_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_1076_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(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% mult_pos_pos
tff(fact_1077_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_1078_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_1079_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_1080_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(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat)) ) ) ).

% mult_pos_neg2
tff(fact_1081_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_1082_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_1083_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_1084_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(aa_rat_fun_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_1085_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_1086_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_1087_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(aa_rat_fun_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_1088_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_1089_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_1090_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_1091_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(aa_rat_fun_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_1092_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_1093_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_1094_mult__less__cancel__left__neg,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1095_mult__less__cancel__left__neg,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1096_mult__less__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1097_mult__less__cancel__left__pos,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1098_mult__less__cancel__left__pos,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1099_mult__less__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1100_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1101_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C2: 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,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1102_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C2: 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,C2),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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1103_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C2: 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),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))) ) ) ).

% mult_strict_left_mono
tff(fact_1104_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))) ) ) ).

% mult_strict_left_mono
tff(fact_1105_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))) ) ) ).

% mult_strict_left_mono
tff(fact_1106_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))) ) ) ).

% mult_strict_left_mono
tff(fact_1107_mult__less__cancel__left__disj,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1108_mult__less__cancel__left__disj,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
          & 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,C2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1109_mult__less__cancel__left__disj,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
          & 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,C2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1110_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1111_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: 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,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1112_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C2: 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,C2),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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1113_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C2: 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),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))) ) ) ).

% mult_strict_right_mono
tff(fact_1114_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))) ) ) ).

% mult_strict_right_mono
tff(fact_1115_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))) ) ) ).

% mult_strict_right_mono
tff(fact_1116_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))) ) ) ).

% mult_strict_right_mono
tff(fact_1117_mult__less__cancel__right__disj,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1118_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
          & 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,C2),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1119_mult__less__cancel__right__disj,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
          & 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,C2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1120_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C2: 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),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1121_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1122_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1123_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1124_not__one__le__zero,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),zero_z3403309356797280102nteger)) ).

% not_one_le_zero
tff(fact_1125_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_1126_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_1127_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_1128_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_1129_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),one_one_Code_integer)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1130_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_1131_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_1132_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_1133_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_1134_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),one_one_Code_integer)) ).

% zero_less_one_class.zero_le_one
tff(fact_1135_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_1136_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_1137_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_1138_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_1139_nat__mult__le__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_1140_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N)))
    <=> ( ( M2 = zero_zero_nat )
        | ? [J: nat] :
            ( ( M2 = aa_nat_nat(suc,J) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_1141_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ? [M: nat] : N = aa_nat_nat(suc,M) ) ).

% gr0_implies_Suc
tff(fact_1142_All__less__Suc2,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P2,I)) )
    <=> ( pp(aa_nat_bool(P2,zero_zero_nat))
        & ! [I: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
           => pp(aa_nat_bool(P2,aa_nat_nat(suc,I))) ) ) ) ).

% All_less_Suc2
tff(fact_1143_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ? [M6: nat] : N = aa_nat_nat(suc,M6) ) ).

% gr0_conv_Suc
tff(fact_1144_Ex__less__Suc2,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P2,I)) )
    <=> ( pp(aa_nat_bool(P2,zero_zero_nat))
        | ? [I: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
            & pp(aa_nat_bool(P2,aa_nat_nat(suc,I))) ) ) ) ).

% Ex_less_Suc2
tff(fact_1145_le__imp__less__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N))) ) ).

% le_imp_less_Suc
tff(fact_1146_less__eq__Suc__le,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M2)) ) ).

% less_eq_Suc_le
tff(fact_1147_less__Suc__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% less_Suc_eq_le
tff(fact_1148_le__less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M2)))
      <=> ( N = M2 ) ) ) ).

% le_less_Suc_eq
tff(fact_1149_Suc__le__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_le_lessD
tff(fact_1150_inc__induct,axiom,
    ! [I2: nat,J2: nat,P2: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
     => ( pp(aa_nat_bool(P2,J2))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),J2))
               => ( pp(aa_nat_bool(P2,aa_nat_nat(suc,N3)))
                 => pp(aa_nat_bool(P2,N3)) ) ) )
         => pp(aa_nat_bool(P2,I2)) ) ) ) ).

% inc_induct
tff(fact_1151_dec__induct,axiom,
    ! [I2: nat,J2: nat,P2: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
     => ( pp(aa_nat_bool(P2,I2))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),J2))
               => ( pp(aa_nat_bool(P2,N3))
                 => pp(aa_nat_bool(P2,aa_nat_nat(suc,N3))) ) ) )
         => pp(aa_nat_bool(P2,J2)) ) ) ) ).

% dec_induct
tff(fact_1152_Suc__le__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_le_eq
tff(fact_1153_Suc__leI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N)) ) ).

% Suc_leI
tff(fact_1154_ex__least__nat__le,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P2,N))
     => ( ~ pp(aa_nat_bool(P2,zero_zero_nat))
       => ? [K: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
               => ~ pp(aa_nat_bool(P2,I4)) )
            & pp(aa_nat_bool(P2,K)) ) ) ) ).

% ex_least_nat_le
tff(fact_1155_Suc__mult__less__cancel1,axiom,
    ! [K2: nat,M2: nat,N: 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,K2)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_1156_mult__less__mono1,axiom,
    ! [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,zero_zero_nat),K2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),K2))) ) ) ).

% mult_less_mono1
tff(fact_1157_mult__less__mono2,axiom,
    ! [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,zero_zero_nat),K2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),I2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),J2))) ) ) ).

% mult_less_mono2
tff(fact_1158_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_1159_mult__eq__self__implies__10,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) )
     => ( ( N = one_one_nat )
        | ( M2 = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_1160_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C2: 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,C2),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),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1161_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C2: 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,C2),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),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1162_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C2: 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,C2),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),C2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1163_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C2: 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,C2),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),C2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1164_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C2: 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,C2),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),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1165_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C2: 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,C2),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),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1166_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C2: 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,C2),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),C2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1167_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C2: 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,C2),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),C2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1168_mult__right__le__imp__le,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_1169_mult__right__le__imp__le,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_1170_mult__right__le__imp__le,axiom,
    ! [A: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_1171_mult__right__le__imp__le,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_1172_mult__left__le__imp__le,axiom,
    ! [C2: 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,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_1173_mult__left__le__imp__le,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_1174_mult__left__le__imp__le,axiom,
    ! [C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_1175_mult__left__le__imp__le,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_1176_mult__le__cancel__left__pos,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1177_mult__le__cancel__left__pos,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1178_mult__le__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1179_mult__le__cancel__left__neg,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1180_mult__le__cancel__left__neg,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1181_mult__le__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1182_mult__less__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_1183_mult__less__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => 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,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_1184_mult__less__cancel__right,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => 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,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_1185_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C2: 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,C2),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),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1186_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C2: 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,C2),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),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1187_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C2: 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,C2),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),C2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1188_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C2: 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,C2),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),C2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1189_mult__right__less__imp__less,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_1190_mult__right__less__imp__less,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_1191_mult__right__less__imp__less,axiom,
    ! [A: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_1192_mult__right__less__imp__less,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_1193_mult__less__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_1194_mult__less__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
         => 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,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_1195_mult__less__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
         => 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,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_1196_mult__strict__mono,axiom,
    ! [A: real,B: real,C2: 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,C2),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),C2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_1197_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C2: 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,C2),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),C2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_1198_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C2: 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,C2),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),C2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_1199_mult__strict__mono,axiom,
    ! [A: int,B: int,C2: 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,C2),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),C2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_1200_mult__left__less__imp__less,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_1201_mult__left__less__imp__less,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_1202_mult__left__less__imp__less,axiom,
    ! [C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_1203_mult__left__less__imp__less,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_1204_mult__le__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_1205_mult__le__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => 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,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_1206_mult__le__cancel__right,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => 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,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_1207_mult__le__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_1208_mult__le__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => 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,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_1209_mult__le__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C2))
         => 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,C2),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_1210_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X3)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw),Ux)
       => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,S),X3) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_1211_mult__left__le,axiom,
    ! [C2: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C2),one_one_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,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),A)) ) ) ).

% mult_left_le
tff(fact_1212_mult__left__le,axiom,
    ! [C2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),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),C2)),A)) ) ) ).

% mult_left_le
tff(fact_1213_mult__left__le,axiom,
    ! [C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),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(aa_rat_fun_rat_rat(times_times_rat,A),C2)),A)) ) ) ).

% mult_left_le
tff(fact_1214_mult__left__le,axiom,
    ! [C2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),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),C2)),A)) ) ) ).

% mult_left_le
tff(fact_1215_mult__left__le,axiom,
    ! [C2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),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),C2)),A)) ) ) ).

% mult_left_le
tff(fact_1216_mult__le__one,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),one_one_Code_integer))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer)) ) ) ) ).

% mult_le_one
tff(fact_1217_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_1218_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(aa_rat_fun_rat_rat(times_times_rat,A),B)),one_one_rat)) ) ) ) ).

% mult_le_one
tff(fact_1219_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_1220_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_1221_mult__right__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),one_one_Code_integer))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_1222_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_1223_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(aa_rat_fun_rat_rat(times_times_rat,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_1224_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_1225_mult__left__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),one_one_Code_integer))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_1226_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_1227_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(aa_rat_fun_rat_rat(times_times_rat,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_1228_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_1229_ex__least__nat__less,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P2,N))
     => ( ~ pp(aa_nat_bool(P2,zero_zero_nat))
       => ? [K: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),N))
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),K))
               => ~ pp(aa_nat_bool(P2,I4)) )
            & pp(aa_nat_bool(P2,aa_nat_nat(suc,K))) ) ) ) ).

% ex_least_nat_less
tff(fact_1230_n__less__n__mult__m,axiom,
    ! [N: nat,M2: 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(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M2))) ) ) ).

% n_less_n_mult_m
tff(fact_1231_n__less__m__mult__n,axiom,
    ! [N: nat,M2: 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(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N))) ) ) ).

% n_less_m_mult_n
tff(fact_1232_one__less__mult,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2))
       => 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,M2),N))) ) ) ).

% one_less_mult
tff(fact_1233_greater__shift,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(X)),some_nat(Y))) ) ).

% greater_shift
tff(fact_1234_less__shift,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(X)),some_nat(Y))) ) ).

% less_shift
tff(fact_1235_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_buildup(N)),N)) ) ).

% buildup_gives_valid
tff(fact_1236_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_1237_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(aa_rat_fun_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_1238_case4_I2_J,axiom,
    ! [S2: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(S2),m))
     => ( ( vEBT_VEBT_set_vebt(summary2) = vEBT_VEBT_set_vebt(S2) )
       => ( S2 = summary2 ) ) ) ).

% case4(2)
tff(fact_1239_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( 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_1240_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( 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_1241_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( 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_1242_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( 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_1243_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_1244_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(aa_rat_fun_rat_rat(times_times_rat,Z2),X)),aa_rat_rat(aa_rat_fun_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_1245_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_1246_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_1247_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(aa_rat_fun_rat_rat(times_times_rat,X),Z2)),aa_rat_rat(aa_rat_fun_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_1248_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_1249_ac,axiom,
    ! [T: vEBT_VEBT,H: nat,K2: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),H))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(K2),H))
       => ( ( vEBT_VEBT_set_vebt(T) = vEBT_VEBT_set_vebt(K2) )
         => ( vEBT_vebt_mint(T) = vEBT_vebt_mint(K2) ) ) ) ) ).

% ac
tff(fact_1250_ad,axiom,
    ! [T: vEBT_VEBT,H: nat,K2: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),H))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(K2),H))
       => ( ( vEBT_VEBT_set_vebt(T) = vEBT_VEBT_set_vebt(K2) )
         => ( vEBT_vebt_maxt(T) = vEBT_vebt_maxt(K2) ) ) ) ) ).

% ad
tff(fact_1251_case4_I5_J,axiom,
    m = na ).

% case4(5)
tff(fact_1252_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(1)
tff(fact_1253_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_1254_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_1255_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(aa_rat_fun_rat_rat(times_times_rat,X),Z2)),aa_rat_rat(aa_rat_fun_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_1256_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_1257_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz)) = some_nat(Ma) ).

% vebt_maxt.simps(3)
tff(fact_1258_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz)) = some_nat(Mi) ).

% vebt_mint.simps(3)
tff(fact_1259_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( 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_1260_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( 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_1261_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_leaf
tff(fact_1262_maxt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => pp(aa_nat_bool(vEBT_vebt_member(T),Maxi)) ) ) ).

% maxt_member
tff(fact_1263_mint__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(Maxi) )
       => pp(aa_nat_bool(vEBT_vebt_member(T),Maxi)) ) ) ).

% mint_member
tff(fact_1264_aa,axiom,
    pp(aa_set_nat_bool(ord_less_eq_set_nat(insert_nat(mi,insert_nat(ma,bot_bot_set_nat))),vEBT_VEBT_set_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList2,summary2)))) ).

% aa
tff(fact_1265_VEBT__internal_Oinsert_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X3)
     => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info,Deg2,TreeList,Summary2),X3) ) ).

% VEBT_internal.insert'.cases
tff(fact_1266_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( 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_1267_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F2: fun_Pr2477239776880934807at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(F2,some_P7363390416028606310at_nat(A),some_P7363390416028606310at_nat(B)) = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(F2,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1268_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F2: fun_num_fun_num_num,A: num,B: num] : vEBT_V819420779217536731ft_num(F2,some_num(A),some_num(B)) = some_num(aa_num_num(aa_num_fun_num_num(F2,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1269_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F2: fun_nat_fun_nat_nat,A: nat,B: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(F2),some_nat(A)),some_nat(B)) = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(F2,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1270_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_1271_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))
        & ! [Z3: nat] :
            ( ( pp(aa_nat_bool(vEBT_vebt_member(T),Z3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z3),Y)) ) ) ) ).

% pred_member
tff(fact_1272_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))
        & ! [Z3: nat] :
            ( ( pp(aa_nat_bool(vEBT_vebt_member(T),Z3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z3)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z3)) ) ) ) ).

% succ_member
tff(fact_1273_case4_I6_J,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% case4(6)
tff(fact_1274_case4_I1_J,axiom,
    ! [X5: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(treeList2)))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(X5),na))
        & ! [Xa: vEBT_VEBT] :
            ( pp(aa_nat_bool(vEBT_invar_vebt(Xa),na))
           => ( ( vEBT_VEBT_set_vebt(X5) = vEBT_VEBT_set_vebt(Xa) )
             => ( Xa = X5 ) ) ) ) ) ).

% case4(1)
tff(fact_1275_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_1276_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),bot_bot_set_real))
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_1277_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_1278_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_1279_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_1280_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),bot_bot_set_real))
    <=> ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_1281_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_1282_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_1283_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_1284_bot_Oextremum,axiom,
    ! [A: set_real] : pp(aa_set_real_bool(ord_less_eq_set_real(bot_bot_set_real),A)) ).

% bot.extremum
tff(fact_1285_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_1286_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_1287_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_1288_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_1289_bot_Oextremum__strict,axiom,
    ! [A: set_real] : ~ pp(aa_set_real_bool(ord_less_set_real(A),bot_bot_set_real)) ).

% bot.extremum_strict
tff(fact_1290_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_1291_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_1292_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_1293_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_set_real(bot_bot_set_real),A)) ) ).

% bot.not_eq_extremum
tff(fact_1294_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_1295_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero_nat,Uv,Uw2)),Ux2)) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_1296_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_1297_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N))
     => ( 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_1298_vebt__member_Osimps_I4_J,axiom,
    ! [V2: 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(V2),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),X)) ).

% vebt_member.simps(4)
tff(fact_1299_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_1300_vebt__member_Osimps_I3_J,axiom,
    ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz)),X)) ).

% vebt_member.simps(3)
tff(fact_1301_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_maxt(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_1302_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_mint(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_1303_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_min_max
tff(fact_1304_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_vebt_delete(T,X)),Y))
      <=> ( ( X != Y )
          & pp(aa_nat_bool(vEBT_vebt_member(T),Y)) ) ) ) ).

% dele_member_cont_corr
tff(fact_1305_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool,Uw: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Uw)
     => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2),X3)
         => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V),TreeList,Vc2),X3)
           => ~ ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd),X3) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_1306_case4_I8_J,axiom,
    ( ( mi = ma )
   => ! [X5: vEBT_VEBT] :
        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(treeList2)))
       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_1)) ) ) ).

% case4(8)
tff(fact_1307_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N3: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3)
     => ~ ! [N3: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) ) ).

% even_odd_cases
tff(fact_1308_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_delete(T,X)),N)) ) ).

% delete_pres_valid
tff(fact_1309_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_1310_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_delete(T,X)),Y))
      <=> ( ( X != Y )
          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),Y)) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_1311_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( 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_1312_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( 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_1313_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_1314_add__right__cancel,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = aa_real_real(plus_plus_real(C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_1315_add__right__cancel,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_1316_add__right__cancel,axiom,
    ! [B: nat,A: nat,C2: 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,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_1317_add__right__cancel,axiom,
    ! [B: int,A: int,C2: 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,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_1318_add__left__cancel,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_1319_add__left__cancel,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_1320_add__left__cancel,axiom,
    ! [A: nat,B: nat,C2: 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),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_1321_add__left__cancel,axiom,
    ! [A: int,B: int,C2: 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),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_1322_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 )
       => ( ! [X5: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList2)))
             => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_1)) )
          & ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_1)) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_1323_add__le__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(C2),A)),aa_real_real(plus_plus_real(C2),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_cancel_left
tff(fact_1324_add__le__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_cancel_left
tff(fact_1325_add__le__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_cancel_left
tff(fact_1326_add__le__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_cancel_left
tff(fact_1327_add__le__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(A),C2)),aa_real_real(plus_plus_real(B),C2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_cancel_right
tff(fact_1328_add__le__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_cancel_right
tff(fact_1329_add__le__cancel__right,axiom,
    ! [A: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_cancel_right
tff(fact_1330_add__le__cancel__right,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_cancel_right
tff(fact_1331_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_1332_add_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% add.right_neutral
tff(fact_1333_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_1334_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_1335_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_1336_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(plus_plus_real(A),A) )
    <=> ( A = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_1337_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_1338_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_1339_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( aa_complex_complex(plus_plus_complex(B),A) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
tff(fact_1340_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_1341_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_1342_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_1343_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_1344_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
tff(fact_1345_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_1346_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_1347_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_1348_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_1349_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
tff(fact_1350_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(plus_plus_real(B),A) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_1351_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_1352_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_1353_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_1354_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(A),B) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
tff(fact_1355_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(plus_plus_real(A),B) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_1356_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_1357_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_1358_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_1359_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_1360_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_1361_add__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add_0
tff(fact_1362_add__0,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% add_0
tff(fact_1363_add__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add_0
tff(fact_1364_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_1365_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_1366_add__less__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),C2)),aa_real_real(plus_plus_real(B),C2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_cancel_right
tff(fact_1367_add__less__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_cancel_right
tff(fact_1368_add__less__cancel__right,axiom,
    ! [A: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_cancel_right
tff(fact_1369_add__less__cancel__right,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_cancel_right
tff(fact_1370_add__less__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(C2),A)),aa_real_real(plus_plus_real(C2),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_cancel_left
tff(fact_1371_add__less__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_cancel_left
tff(fact_1372_add__less__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_cancel_left
tff(fact_1373_add__less__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_cancel_left
tff(fact_1374_add__Suc__right,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) ).

% add_Suc_right
tff(fact_1375_Nat_Oadd__0__right,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),zero_zero_nat) = M2 ).

% Nat.add_0_right
tff(fact_1376_add__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N) = zero_zero_nat )
    <=> ( ( M2 = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_1377_nat__add__left__cancel__less,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% nat_add_left_cancel_less
tff(fact_1378_nat__add__left__cancel__le,axiom,
    ! [K2: nat,M2: nat,N: 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,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% nat_add_left_cancel_le
tff(fact_1379_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ! [Y5: product_prod_nat_nat] : X != some_P7363390416028606310at_nat(Y5)
    <=> ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_1380_not__Some__eq,axiom,
    ! [X: option_nat] :
      ( ! [Y5: nat] : X != some_nat(Y5)
    <=> ( X = none_nat ) ) ).

% not_Some_eq
tff(fact_1381_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ! [Y5: num] : X != some_num(Y5)
    <=> ( X = none_num ) ) ).

% not_Some_eq
tff(fact_1382_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
    <=> ? [Y5: product_prod_nat_nat] : X = some_P7363390416028606310at_nat(Y5) ) ).

% not_None_eq
tff(fact_1383_not__None__eq,axiom,
    ! [X: option_nat] :
      ( ( X != none_nat )
    <=> ? [Y5: nat] : X = some_nat(Y5) ) ).

% not_None_eq
tff(fact_1384_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
    <=> ? [Y5: num] : X = some_num(Y5) ) ).

% not_None_eq
tff(fact_1385_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_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_1386_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(aa_rat_fun_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_1387_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_1388_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_1389_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_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_1390_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(aa_rat_fun_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_1391_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_1392_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_1393_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_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_1394_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(aa_rat_fun_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_1395_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_1396_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_1397_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_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_1398_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(aa_rat_fun_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_1399_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_1400_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_1401_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(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_1402_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(aa_rat_fun_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_1403_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_1404_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(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_1405_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(aa_rat_fun_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_1406_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_1407_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(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_1408_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(aa_rat_fun_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_1409_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_1410_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(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_1411_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(aa_rat_fun_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_1412_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_1413_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_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_1414_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(aa_rat_fun_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_1415_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_1416_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_1417_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_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_1418_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(aa_rat_fun_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_1419_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_1420_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_1421_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_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_1422_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_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_1423_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_1424_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_1425_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_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_1426_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_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_1427_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_1428_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_1429_add__gr__0,axiom,
    ! [M2: nat,N: 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,M2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% add_gr_0
tff(fact_1430_mult__Suc__right,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) ).

% mult_Suc_right
tff(fact_1431_add__right__imp__eq,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = aa_real_real(plus_plus_real(C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_1432_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_1433_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C2: 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,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_1434_add__right__imp__eq,axiom,
    ! [B: int,A: int,C2: 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,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_1435_add__left__imp__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_1436_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_1437_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C2: 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),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_1438_add__left__imp__eq,axiom,
    ! [A: int,B: int,C2: 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),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_1439_add_Oleft__commute,axiom,
    ! [B: real,A: real,C2: real] : aa_real_real(plus_plus_real(B),aa_real_real(plus_plus_real(A),C2)) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C2)) ).

% add.left_commute
tff(fact_1440_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) ).

% add.left_commute
tff(fact_1441_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: 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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% add.left_commute
tff(fact_1442_add_Oleft__commute,axiom,
    ! [B: int,A: int,C2: 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),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% add.left_commute
tff(fact_1443_add_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(B),A) ).

% add.commute
tff(fact_1444_add_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) ).

% add.commute
tff(fact_1445_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_1446_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_1447_add_Oright__cancel,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = aa_real_real(plus_plus_real(C2),A) )
    <=> ( B = C2 ) ) ).

% add.right_cancel
tff(fact_1448_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A) )
    <=> ( B = C2 ) ) ).

% add.right_cancel
tff(fact_1449_add_Oright__cancel,axiom,
    ! [B: int,A: int,C2: 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,C2),A) )
    <=> ( B = C2 ) ) ).

% add.right_cancel
tff(fact_1450_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(A),C2) )
    <=> ( B = C2 ) ) ).

% add.left_cancel
tff(fact_1451_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2) )
    <=> ( B = C2 ) ) ).

% add.left_cancel
tff(fact_1452_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C2: 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),C2) )
    <=> ( B = C2 ) ) ).

% add.left_cancel
tff(fact_1453_add_Oassoc,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B)),C2) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C2)) ).

% add.assoc
tff(fact_1454_add_Oassoc,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) ).

% add.assoc
tff(fact_1455_add_Oassoc,axiom,
    ! [A: nat,B: nat,C2: 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)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% add.assoc
tff(fact_1456_add_Oassoc,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% add.assoc
tff(fact_1457_group__cancel_Oadd2,axiom,
    ! [B5: real,K2: real,B: real,A: real] :
      ( ( B5 = aa_real_real(plus_plus_real(K2),B) )
     => ( aa_real_real(plus_plus_real(A),B5) = aa_real_real(plus_plus_real(K2),aa_real_real(plus_plus_real(A),B)) ) ) ).

% group_cancel.add2
tff(fact_1458_group__cancel_Oadd2,axiom,
    ! [B5: rat,K2: rat,B: rat,A: rat] :
      ( ( B5 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B5) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_1459_group__cancel_Oadd2,axiom,
    ! [B5: nat,K2: nat,B: nat,A: nat] :
      ( ( B5 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_1460_group__cancel_Oadd2,axiom,
    ! [B5: int,K2: int,B: int,A: int] :
      ( ( B5 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B5) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_1461_group__cancel_Oadd1,axiom,
    ! [A4: real,K2: real,A: real,B: real] :
      ( ( A4 = aa_real_real(plus_plus_real(K2),A) )
     => ( aa_real_real(plus_plus_real(A4),B) = aa_real_real(plus_plus_real(K2),aa_real_real(plus_plus_real(A),B)) ) ) ).

% group_cancel.add1
tff(fact_1462_group__cancel_Oadd1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A4),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_1463_group__cancel_Oadd1,axiom,
    ! [A4: nat,K2: nat,A: nat,B: nat] :
      ( ( A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_1464_group__cancel_Oadd1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A4),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_1465_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( ( I2 = J2 )
        & ( K2 = L ) )
     => ( aa_real_real(plus_plus_real(I2),K2) = aa_real_real(plus_plus_real(J2),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1466_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( I2 = J2 )
        & ( K2 = L ) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1467_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( I2 = J2 )
        & ( K2 = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1468_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( I2 = J2 )
        & ( K2 = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1469_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B)),C2) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C2)) ).

% is_num_normalize(1)
tff(fact_1470_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) ).

% is_num_normalize(1)
tff(fact_1471_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% is_num_normalize(1)
tff(fact_1472_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B)),C2) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C2)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_1473_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_1474_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C2: 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)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_1475_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_1476_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_1477_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)),Uz)) ).

% VEBT_internal.membermima.simps(2)
tff(fact_1478_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I2),J2))
        & ( K2 = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(I2),K2)),aa_real_real(plus_plus_real(J2),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1479_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I2),J2))
        & ( K2 = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1480_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
        & ( K2 = 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,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1481_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J2))
        & ( K2 = 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,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1482_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( ( I2 = J2 )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K2),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(I2),K2)),aa_real_real(plus_plus_real(J2),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1483_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( I2 = J2 )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K2),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1484_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( I2 = J2 )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),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,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1485_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( I2 = J2 )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),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,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1486_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I2),J2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K2),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(I2),K2)),aa_real_real(plus_plus_real(J2),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1487_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I2),J2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K2),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1488_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: 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_eq_nat,K2),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,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1489_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),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,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1490_add__mono,axiom,
    ! [A: real,B: real,C2: 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,C2),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(A),C2)),aa_real_real(plus_plus_real(B),D))) ) ) ).

% add_mono
tff(fact_1491_add__mono,axiom,
    ! [A: rat,B: rat,C2: 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,C2),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D))) ) ) ).

% add_mono
tff(fact_1492_add__mono,axiom,
    ! [A: nat,B: nat,C2: 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,C2),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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_mono
tff(fact_1493_add__mono,axiom,
    ! [A: int,B: int,C2: 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,C2),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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_mono
tff(fact_1494_add__left__mono,axiom,
    ! [A: real,B: real,C2: 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(plus_plus_real(C2),A)),aa_real_real(plus_plus_real(C2),B))) ) ).

% add_left_mono
tff(fact_1495_add__left__mono,axiom,
    ! [A: rat,B: rat,C2: 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(aa_rat_fun_rat_rat(plus_plus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B))) ) ).

% add_left_mono
tff(fact_1496_add__left__mono,axiom,
    ! [A: nat,B: nat,C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))) ) ).

% add_left_mono
tff(fact_1497_add__left__mono,axiom,
    ! [A: int,B: int,C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))) ) ).

% add_left_mono
tff(fact_1498_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ~ ! [C: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ) ).

% less_eqE
tff(fact_1499_add__right__mono,axiom,
    ! [A: real,B: real,C2: 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(plus_plus_real(A),C2)),aa_real_real(plus_plus_real(B),C2))) ) ).

% add_right_mono
tff(fact_1500_add__right__mono,axiom,
    ! [A: rat,B: rat,C2: 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(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2))) ) ).

% add_right_mono
tff(fact_1501_add__right__mono,axiom,
    ! [A: nat,B: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))) ) ).

% add_right_mono
tff(fact_1502_add__right__mono,axiom,
    ! [A: int,B: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))) ) ).

% add_right_mono
tff(fact_1503_le__iff__add,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) ) ).

% le_iff_add
tff(fact_1504_add__le__imp__le__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(C2),A)),aa_real_real(plus_plus_real(C2),B)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_imp_le_left
tff(fact_1505_add__le__imp__le__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_imp_le_left
tff(fact_1506_add__le__imp__le__left,axiom,
    ! [C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_imp_le_left
tff(fact_1507_add__le__imp__le__left,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_imp_le_left
tff(fact_1508_add__le__imp__le__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(A),C2)),aa_real_real(plus_plus_real(B),C2)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_imp_le_right
tff(fact_1509_add__le__imp__le__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_imp_le_right
tff(fact_1510_add__le__imp__le__right,axiom,
    ! [A: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_imp_le_right
tff(fact_1511_add__le__imp__le__right,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_imp_le_right
tff(fact_1512_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1513_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1514_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1515_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_1516_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_1517_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_1518_add_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% add.comm_neutral
tff(fact_1519_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_1520_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_1521_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_1522_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_1523_add_Ogroup__left__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% add.group_left_neutral
tff(fact_1524_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_1525_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_1526_add__less__imp__less__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),C2)),aa_real_real(plus_plus_real(B),C2)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_imp_less_right
tff(fact_1527_add__less__imp__less__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_imp_less_right
tff(fact_1528_add__less__imp__less__right,axiom,
    ! [A: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_imp_less_right
tff(fact_1529_add__less__imp__less__right,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_imp_less_right
tff(fact_1530_add__less__imp__less__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(C2),A)),aa_real_real(plus_plus_real(C2),B)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_imp_less_left
tff(fact_1531_add__less__imp__less__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_imp_less_left
tff(fact_1532_add__less__imp__less__left,axiom,
    ! [C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_imp_less_left
tff(fact_1533_add__less__imp__less__left,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_imp_less_left
tff(fact_1534_add__strict__right__mono,axiom,
    ! [A: real,B: real,C2: 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(plus_plus_real(A),C2)),aa_real_real(plus_plus_real(B),C2))) ) ).

% add_strict_right_mono
tff(fact_1535_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: 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(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2))) ) ).

% add_strict_right_mono
tff(fact_1536_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))) ) ).

% add_strict_right_mono
tff(fact_1537_add__strict__right__mono,axiom,
    ! [A: int,B: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))) ) ).

% add_strict_right_mono
tff(fact_1538_add__strict__left__mono,axiom,
    ! [A: real,B: real,C2: 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(plus_plus_real(C2),A)),aa_real_real(plus_plus_real(C2),B))) ) ).

% add_strict_left_mono
tff(fact_1539_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: 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(aa_rat_fun_rat_rat(plus_plus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B))) ) ).

% add_strict_left_mono
tff(fact_1540_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))) ) ).

% add_strict_left_mono
tff(fact_1541_add__strict__left__mono,axiom,
    ! [A: int,B: int,C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))) ) ).

% add_strict_left_mono
tff(fact_1542_add__strict__mono,axiom,
    ! [A: real,B: real,C2: 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,C2),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),C2)),aa_real_real(plus_plus_real(B),D))) ) ) ).

% add_strict_mono
tff(fact_1543_add__strict__mono,axiom,
    ! [A: rat,B: rat,C2: 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,C2),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D))) ) ) ).

% add_strict_mono
tff(fact_1544_add__strict__mono,axiom,
    ! [A: nat,B: nat,C2: 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,C2),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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_strict_mono
tff(fact_1545_add__strict__mono,axiom,
    ! [A: int,B: int,C2: 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,C2),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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_strict_mono
tff(fact_1546_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I2),J2))
        & ( K2 = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(I2),K2)),aa_real_real(plus_plus_real(J2),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1547_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I2),J2))
        & ( K2 = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1548_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
        & ( K2 = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1549_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),J2))
        & ( K2 = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1550_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( ( I2 = J2 )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K2),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(I2),K2)),aa_real_real(plus_plus_real(J2),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1551_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( I2 = J2 )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K2),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1552_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( I2 = J2 )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1553_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( I2 = J2 )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1554_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I2),J2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K2),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(I2),K2)),aa_real_real(plus_plus_real(J2),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1555_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I2),J2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K2),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1556_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: 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,K2),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1557_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),J2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1558_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(plus_plus_real(A),B)),C2) = aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_1559_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_1560_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_1561_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(plus_plus_real(B),C2)) = aa_real_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),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_1562_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_1563_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C2: 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),C2)) = 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),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_1564_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(plus_plus_real(A),B)),C2) = aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% comm_semiring_class.distrib
tff(fact_1565_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ).

% comm_semiring_class.distrib
tff(fact_1566_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C2: 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)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% comm_semiring_class.distrib
tff(fact_1567_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% comm_semiring_class.distrib
tff(fact_1568_distrib__left,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(plus_plus_real(B),C2)) = aa_real_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),C2)) ).

% distrib_left
tff(fact_1569_distrib__left,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)) ).

% distrib_left
tff(fact_1570_distrib__left,axiom,
    ! [A: nat,B: nat,C2: 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),C2)) = 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),C2)) ).

% distrib_left
tff(fact_1571_distrib__left,axiom,
    ! [A: int,B: int,C2: 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),C2)) = 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),C2)) ).

% distrib_left
tff(fact_1572_distrib__right,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(plus_plus_real(A),B)),C2) = aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% distrib_right
tff(fact_1573_distrib__right,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ).

% distrib_right
tff(fact_1574_distrib__right,axiom,
    ! [A: nat,B: nat,C2: 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)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% distrib_right
tff(fact_1575_distrib__right,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% distrib_right
tff(fact_1576_combine__common__factor,axiom,
    ! [A: real,E: real,B: real,C2: real] : aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),C2)) = aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(plus_plus_real(A),B)),E)),C2) ).

% combine_common_factor
tff(fact_1577_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),E)),C2) ).

% combine_common_factor
tff(fact_1578_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E)),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E)),C2) ).

% combine_common_factor
tff(fact_1579_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E)),C2) ).

% combine_common_factor
tff(fact_1580_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P2: fun_op5682630191948829869t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P2,X),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P2,X),Y)) )
       => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( X = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P2,X),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P2,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_1581_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P2: fun_op9100744243108201484t_bool,Y: option_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P2,X),Y)) )
     => ( ( ( Y = none_nat )
         => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P2,X),Y)) )
       => ( ! [A3: product_prod_nat_nat,B3: nat] :
              ( ( X = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_nat(B3) )
               => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P2,X),Y)) ) )
         => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P2,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_1582_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P2: fun_op775672374849087042m_bool,Y: option_num] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P2,X),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P2,X),Y)) )
       => ( ! [A3: product_prod_nat_nat,B3: num] :
              ( ( X = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_num(B3) )
               => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P2,X),Y)) ) )
         => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P2,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_1583_combine__options__cases,axiom,
    ! [X: option_nat,P2: fun_op5698422665285572208t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_nat )
       => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P2,X),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P2,X),Y)) )
       => ( ! [A3: nat,B3: product_prod_nat_nat] :
              ( ( X = some_nat(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P2,X),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P2,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_1584_combine__options__cases,axiom,
    ! [X: option_nat,P2: fun_op7792301606653096457t_bool,Y: option_nat] :
      ( ( ( X = none_nat )
       => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P2,X),Y)) )
     => ( ( ( Y = none_nat )
         => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P2,X),Y)) )
       => ( ! [A3: nat,B3: nat] :
              ( ( X = some_nat(A3) )
             => ( ( Y = some_nat(B3) )
               => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P2,X),Y)) ) )
         => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P2,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_1585_combine__options__cases,axiom,
    ! [X: option_nat,P2: fun_op8690601775248757823m_bool,Y: option_num] :
      ( ( ( X = none_nat )
       => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P2,X),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P2,X),Y)) )
       => ( ! [A3: nat,B3: num] :
              ( ( X = some_nat(A3) )
             => ( ( Y = some_num(B3) )
               => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P2,X),Y)) ) )
         => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P2,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_1586_combine__options__cases,axiom,
    ! [X: option_num,P2: fun_op7732454363416259258t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_num )
       => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P2,X),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P2,X),Y)) )
       => ( ! [A3: num,B3: product_prod_nat_nat] :
              ( ( X = some_num(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P2,X),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P2,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_1587_combine__options__cases,axiom,
    ! [X: option_num,P2: fun_op1064277265382982207t_bool,Y: option_nat] :
      ( ( ( X = none_num )
       => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P2,X),Y)) )
     => ( ( ( Y = none_nat )
         => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P2,X),Y)) )
       => ( ! [A3: num,B3: nat] :
              ( ( X = some_num(A3) )
             => ( ( Y = some_nat(B3) )
               => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P2,X),Y)) ) )
         => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P2,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_1588_combine__options__cases,axiom,
    ! [X: option_num,P2: fun_op1962577433978643573m_bool,Y: option_num] :
      ( ( ( X = none_num )
       => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P2,X),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P2,X),Y)) )
       => ( ! [A3: num,B3: num] :
              ( ( X = some_num(A3) )
             => ( ( Y = some_num(B3) )
               => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P2,X),Y)) ) )
         => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P2,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_1589_split__option__all,axiom,
    ! [P2: fun_op2627489687811228755t_bool] :
      ( ! [X_13: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P2,X_13))
    <=> ( pp(aa_opt6800557945374683724t_bool(P2,none_P5556105721700978146at_nat))
        & ! [X4: product_prod_nat_nat] : pp(aa_opt6800557945374683724t_bool(P2,some_P7363390416028606310at_nat(X4))) ) ) ).

% split_option_all
tff(fact_1590_split__option__all,axiom,
    ! [P2: fun_option_nat_bool] :
      ( ! [X_13: option_nat] : pp(aa_option_nat_bool(P2,X_13))
    <=> ( pp(aa_option_nat_bool(P2,none_nat))
        & ! [X4: nat] : pp(aa_option_nat_bool(P2,some_nat(X4))) ) ) ).

% split_option_all
tff(fact_1591_split__option__all,axiom,
    ! [P2: fun_option_num_bool] :
      ( ! [X_13: option_num] : pp(aa_option_num_bool(P2,X_13))
    <=> ( pp(aa_option_num_bool(P2,none_num))
        & ! [X4: num] : pp(aa_option_num_bool(P2,some_num(X4))) ) ) ).

% split_option_all
tff(fact_1592_split__option__ex,axiom,
    ! [P2: fun_op2627489687811228755t_bool] :
      ( ? [X_13: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P2,X_13))
    <=> ( pp(aa_opt6800557945374683724t_bool(P2,none_P5556105721700978146at_nat))
        | ? [X4: product_prod_nat_nat] : pp(aa_opt6800557945374683724t_bool(P2,some_P7363390416028606310at_nat(X4))) ) ) ).

% split_option_ex
tff(fact_1593_split__option__ex,axiom,
    ! [P2: fun_option_nat_bool] :
      ( ? [X_13: option_nat] : pp(aa_option_nat_bool(P2,X_13))
    <=> ( pp(aa_option_nat_bool(P2,none_nat))
        | ? [X4: nat] : pp(aa_option_nat_bool(P2,some_nat(X4))) ) ) ).

% split_option_ex
tff(fact_1594_split__option__ex,axiom,
    ! [P2: fun_option_num_bool] :
      ( ? [X_13: option_num] : pp(aa_option_num_bool(P2,X_13))
    <=> ( pp(aa_option_num_bool(P2,none_num))
        | ? [X4: num] : pp(aa_option_num_bool(P2,some_num(X4))) ) ) ).

% split_option_ex
tff(fact_1595_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X23) ) ).

% option.exhaust
tff(fact_1596_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] : Y != some_nat(X23) ) ).

% option.exhaust
tff(fact_1597_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] : Y != some_num(X23) ) ).

% option.exhaust
tff(fact_1598_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X2) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_1599_option_OdiscI,axiom,
    ! [Option: option_nat,X2: nat] :
      ( ( Option = some_nat(X2) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_1600_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option = some_num(X2) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_1601_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X2) ).

% option.distinct(1)
tff(fact_1602_option_Odistinct_I1_J,axiom,
    ! [X2: nat] : none_nat != some_nat(X2) ).

% option.distinct(1)
tff(fact_1603_option_Odistinct_I1_J,axiom,
    ! [X2: num] : none_num != some_num(X2) ).

% option.distinct(1)
tff(fact_1604_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc1260151622200656695at_nat] :
      ( ! [Uu2: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : X != produc7814092408141220007at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw: fun_Pr2477239776880934807at_nat,V: product_prod_nat_nat] : X != produc7814092408141220007at_nat(Uw,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V),none_P5556105721700978146at_nat))
       => ~ ! [F: fun_Pr2477239776880934807at_nat,A3: product_prod_nat_nat,B3: product_prod_nat_nat] : X != produc7814092408141220007at_nat(F,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A3),some_P7363390416028606310at_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1605_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc7004362771917097900on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_nat,Uv2: option_nat] : X != produc6699827698519632094on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw: fun_nat_fun_nat_nat,V: nat] : X != produc6699827698519632094on_nat(Uw,produc5098337634421038937on_nat(some_nat(V),none_nat))
       => ~ ! [F: fun_nat_fun_nat_nat,A3: nat,B3: nat] : X != produc6699827698519632094on_nat(F,produc5098337634421038937on_nat(some_nat(A3),some_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1606_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc2703417854250991842on_num] :
      ( ! [Uu2: fun_num_fun_num_num,Uv2: option_num] : X != produc6360833703471310868on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw: fun_num_fun_num_num,V: num] : X != produc6360833703471310868on_num(Uw,produc8585076106096196333on_num(some_num(V),none_num))
       => ~ ! [F: fun_num_fun_num_num,A3: num,B3: num] : X != produc6360833703471310868on_num(F,produc8585076106096196333on_num(some_num(A3),some_num(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1607_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc533602882674032673at_nat] :
      ( ! [Uu2: fun_Pr248683899324676269t_bool,Uv2: option4927543243414619207at_nat] : X != produc5961576194287291153at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw: fun_Pr248683899324676269t_bool,V: product_prod_nat_nat] : X != produc5961576194287291153at_nat(Uw,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V),none_P5556105721700978146at_nat))
       => ~ ! [F: fun_Pr248683899324676269t_bool,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] : X != produc5961576194287291153at_nat(F,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X3),some_P7363390416028606310at_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1608_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc3700678068921409119on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_bool,Uv2: option_nat] : X != produc2004789812759622991on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw: fun_nat_fun_nat_bool,V: nat] : X != produc2004789812759622991on_nat(Uw,produc5098337634421038937on_nat(some_nat(V),none_nat))
       => ~ ! [F: fun_nat_fun_nat_bool,X3: nat,Y3: nat] : X != produc2004789812759622991on_nat(F,produc5098337634421038937on_nat(some_nat(X3),some_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1609_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc6167219322466932359on_num] :
      ( ! [Uu2: fun_num_fun_num_bool,Uv2: option_num] : X != produc8433281988922931063on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw: fun_num_fun_num_bool,V: num] : X != produc8433281988922931063on_num(Uw,produc8585076106096196333on_num(some_num(V),none_num))
       => ~ ! [F: fun_num_fun_num_bool,X3: num,Y3: num] : X != produc8433281988922931063on_num(F,produc8585076106096196333on_num(some_num(X3),some_num(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1610_nat__arith_Osuc1,axiom,
    ! [A4: nat,K2: nat,A: nat] :
      ( ( A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),A) )
     => ( aa_nat_nat(suc,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_1611_add__Suc,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M2)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) ).

% add_Suc
tff(fact_1612_add__Suc__shift,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M2)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(suc,N)) ).

% add_Suc_shift
tff(fact_1613_add__eq__self__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N) = M2 )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_1614_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N ).

% plus_nat.add_0
tff(fact_1615_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),L))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% less_add_eq_less
tff(fact_1616_trans__less__add2,axiom,
    ! [I2: nat,J2: nat,M2: 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,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),J2))) ) ).

% trans_less_add2
tff(fact_1617_trans__less__add1,axiom,
    ! [I2: nat,J2: nat,M2: 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,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),M2))) ) ).

% trans_less_add1
tff(fact_1618_add__less__mono1,axiom,
    ! [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,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2))) ) ).

% add_less_mono1
tff(fact_1619_not__add__less2,axiom,
    ! [J2: nat,I2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),I2)),I2)) ).

% not_add_less2
tff(fact_1620_not__add__less1,axiom,
    ! [I2: nat,J2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J2)),I2)) ).

% not_add_less1
tff(fact_1621_add__less__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: 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,K2),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ) ).

% add_less_mono
tff(fact_1622_add__lessD1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J2)),K2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K2)) ) ).

% add_lessD1
tff(fact_1623_add__leE,axiom,
    ! [M2: nat,K2: nat,N: 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,M2),K2)),N))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N)) ) ) ).

% add_leE
tff(fact_1624_le__add1,axiom,
    ! [N: nat,M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2))) ).

% le_add1
tff(fact_1625_le__add2,axiom,
    ! [N: nat,M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) ).

% le_add2
tff(fact_1626_add__leD1,axiom,
    ! [M2: nat,K2: nat,N: 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,M2),K2)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% add_leD1
tff(fact_1627_add__leD2,axiom,
    ! [M2: nat,K2: nat,N: 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,M2),K2)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N)) ) ).

% add_leD2
tff(fact_1628_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),L))
     => ? [N3: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N3) ) ).

% le_Suc_ex
tff(fact_1629_add__le__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: 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_eq_nat,K2),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,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ) ).

% add_le_mono
tff(fact_1630_add__le__mono1,axiom,
    ! [I2: nat,J2: nat,K2: 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_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2))) ) ).

% add_le_mono1
tff(fact_1631_trans__le__add1,axiom,
    ! [I2: nat,J2: nat,M2: 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_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),M2))) ) ).

% trans_le_add1
tff(fact_1632_trans__le__add2,axiom,
    ! [I2: nat,J2: nat,M2: 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_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),J2))) ) ).

% trans_le_add2
tff(fact_1633_nat__le__iff__add,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
    <=> ? [K3: nat] : N = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K3) ) ).

% nat_le_iff_add
tff(fact_1634_add__mult__distrib2,axiom,
    ! [K2: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)) ).

% add_mult_distrib2
tff(fact_1635_add__mult__distrib,axiom,
    ! [M2: nat,N: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K2)) ).

% add_mult_distrib
tff(fact_1636_left__add__mult__distrib,axiom,
    ! [I2: nat,U: nat,J2: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J2)),U)),K2) ).

% left_add_mult_distrib
tff(fact_1637_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,Uw2: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Leaf(Uu,Uv)),Uw2)) ).

% VEBT_internal.membermima.simps(1)
tff(fact_1638_add__decreasing,axiom,
    ! [A: real,C2: 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,C2),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(A),C2)),B)) ) ) ).

% add_decreasing
tff(fact_1639_add__decreasing,axiom,
    ! [A: rat,C2: 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,C2),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),B)) ) ) ).

% add_decreasing
tff(fact_1640_add__decreasing,axiom,
    ! [A: nat,C2: 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,C2),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),C2)),B)) ) ) ).

% add_decreasing
tff(fact_1641_add__decreasing,axiom,
    ! [A: int,C2: 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,C2),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),C2)),B)) ) ) ).

% add_decreasing
tff(fact_1642_add__increasing,axiom,
    ! [A: real,B: real,C2: 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),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(plus_plus_real(A),C2))) ) ) ).

% add_increasing
tff(fact_1643_add__increasing,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2))) ) ) ).

% add_increasing
tff(fact_1644_add__increasing,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => 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),C2))) ) ) ).

% add_increasing
tff(fact_1645_add__increasing,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => 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),C2))) ) ) ).

% add_increasing
tff(fact_1646_add__decreasing2,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),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(plus_plus_real(A),C2)),B)) ) ) ).

% add_decreasing2
tff(fact_1647_add__decreasing2,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),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(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),B)) ) ) ).

% add_decreasing2
tff(fact_1648_add__decreasing2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),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),C2)),B)) ) ) ).

% add_decreasing2
tff(fact_1649_add__decreasing2,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),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),C2)),B)) ) ) ).

% add_decreasing2
tff(fact_1650_add__increasing2,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C2))
     => ( 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(plus_plus_real(A),C2))) ) ) ).

% add_increasing2
tff(fact_1651_add__increasing2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C2))
     => ( 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(aa_rat_fun_rat_rat(plus_plus_rat,A),C2))) ) ) ).

% add_increasing2
tff(fact_1652_add__increasing2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
     => ( 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),C2))) ) ) ).

% add_increasing2
tff(fact_1653_add__increasing2,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
     => ( 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),C2))) ) ) ).

% add_increasing2
tff(fact_1654_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(plus_plus_real(A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_1655_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_1656_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_1657_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_1658_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(plus_plus_real(A),B)),zero_zero_real)) ) ) ).

% add_nonpos_nonpos
tff(fact_1659_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).

% add_nonpos_nonpos
tff(fact_1660_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_1661_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_1662_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(plus_plus_real(X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1663_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(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1664_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_1665_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_1666_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(plus_plus_real(X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1667_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(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1668_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_1669_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_1670_add__less__le__mono,axiom,
    ! [A: real,B: real,C2: 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,C2),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),C2)),aa_real_real(plus_plus_real(B),D))) ) ) ).

% add_less_le_mono
tff(fact_1671_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C2: 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,C2),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D))) ) ) ).

% add_less_le_mono
tff(fact_1672_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C2: 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,C2),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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_less_le_mono
tff(fact_1673_add__less__le__mono,axiom,
    ! [A: int,B: int,C2: 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,C2),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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_less_le_mono
tff(fact_1674_add__le__less__mono,axiom,
    ! [A: real,B: real,C2: 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,C2),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),C2)),aa_real_real(plus_plus_real(B),D))) ) ) ).

% add_le_less_mono
tff(fact_1675_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C2: 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,C2),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D))) ) ) ).

% add_le_less_mono
tff(fact_1676_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C2: 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,C2),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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_le_less_mono
tff(fact_1677_add__le__less__mono,axiom,
    ! [A: int,B: int,C2: 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,C2),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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_le_less_mono
tff(fact_1678_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I2),J2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K2),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(I2),K2)),aa_real_real(plus_plus_real(J2),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1679_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I2),J2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K2),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1680_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: 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_eq_nat,K2),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1681_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),J2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1682_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I2),J2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K2),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(I2),K2)),aa_real_real(plus_plus_real(J2),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1683_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I2),J2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K2),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1684_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: 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,K2),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1685_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1686_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_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_1687_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_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_1688_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_1689_pos__add__strict,axiom,
    ! [A: real,B: real,C2: 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),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(plus_plus_real(A),C2))) ) ) ).

% pos_add_strict
tff(fact_1690_pos__add__strict,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2))) ) ) ).

% pos_add_strict
tff(fact_1691_pos__add__strict,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => 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),C2))) ) ) ).

% pos_add_strict
tff(fact_1692_pos__add__strict,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => 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),C2))) ) ) ).

% pos_add_strict
tff(fact_1693_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))
     => ~ ! [C: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1694_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(plus_plus_real(A),B))) ) ) ).

% add_pos_pos
tff(fact_1695_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).

% add_pos_pos
tff(fact_1696_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_1697_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_1698_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(plus_plus_real(A),B)),zero_zero_real)) ) ) ).

% add_neg_neg
tff(fact_1699_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).

% add_neg_neg
tff(fact_1700_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_1701_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_1702_add__mono1,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,plus_p5714425477246183910nteger(A,one_one_Code_integer)),plus_p5714425477246183910nteger(B,one_one_Code_integer))) ) ).

% add_mono1
tff(fact_1703_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(plus_plus_real(A),one_one_real)),aa_real_real(plus_plus_real(B),one_one_real))) ) ).

% add_mono1
tff(fact_1704_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(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),one_one_rat))) ) ).

% add_mono1
tff(fact_1705_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_1706_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_1707_less__add__one,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),plus_p5714425477246183910nteger(A,one_one_Code_integer))) ).

% less_add_one
tff(fact_1708_less__add__one,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(plus_plus_real(A),one_one_real))) ).

% less_add_one
tff(fact_1709_less__add__one,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat))) ).

% less_add_one
tff(fact_1710_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_1711_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_1712_add__is__1,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M2 = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_1713_one__is__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N) )
    <=> ( ( ( M2 = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_1714_less__natE,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ~ ! [Q3: nat] : N != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Q3)) ) ).

% less_natE
tff(fact_1715_less__add__Suc1,axiom,
    ! [I2: nat,M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),M2)))) ).

% less_add_Suc1
tff(fact_1716_less__add__Suc2,axiom,
    ! [I2: nat,M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),I2)))) ).

% less_add_Suc2
tff(fact_1717_less__iff__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
    <=> ? [K3: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K3)) ) ).

% less_iff_Suc_add
tff(fact_1718_less__imp__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ? [K: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K)) ) ).

% less_imp_Suc_add
tff(fact_1719_less__imp__add__positive,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
     => ? [K: 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(plus_plus_nat,I2),K) = J2 ) ) ) ).

% less_imp_add_positive
tff(fact_1720_mono__nat__linear__lb,axiom,
    ! [F2: fun_nat_nat,M2: nat,K2: nat] :
      ( ! [M: nat,N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F2,M)),aa_nat_nat(F2,N3))) )
     => 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(F2,M2)),K2)),aa_nat_nat(F2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K2)))) ) ).

% mono_nat_linear_lb
tff(fact_1721_mult__Suc,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M2)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) ).

% mult_Suc
tff(fact_1722_Suc__eq__plus1,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ).

% Suc_eq_plus1
tff(fact_1723_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_1724_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ).

% Suc_eq_plus1_left
tff(fact_1725_VEBT__internal_OminNull_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X != vEBT_Leaf(fFalse,fFalse) )
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ( ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy)
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_1726_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(Uw2,some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1727_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_num_fun_num_num,V2: num] : vEBT_V819420779217536731ft_num(Uw2,some_num(V2),none_num) = none_num ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1728_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_nat_fun_nat_nat,V2: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uw2),some_nat(V2)),none_nat) = none_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1729_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_Pr2477239776880934807at_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(X,Xa2,Xb) = Y )
     => ( ( ( Xa2 = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V: product_prod_nat_nat] : Xa2 = some_P7363390416028606310at_nat(V)
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A3: product_prod_nat_nat] :
                ( ( Xa2 = some_P7363390416028606310at_nat(A3) )
               => ! [B3: product_prod_nat_nat] :
                    ( ( Xb = some_P7363390416028606310at_nat(B3) )
                   => ( Y != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(X,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1730_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_num_fun_num_num,Xa2: option_num,Xb: option_num,Y: option_num] :
      ( ( vEBT_V819420779217536731ft_num(X,Xa2,Xb) = Y )
     => ( ( ( Xa2 = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V: num] : Xa2 = some_num(V)
           => ( ( Xb = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A3: num] :
                ( ( Xa2 = some_num(A3) )
               => ! [B3: num] :
                    ( ( Xb = some_num(B3) )
                   => ( Y != some_num(aa_num_num(aa_num_fun_num_num(X,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1731_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa2: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(X),Xa2),Xb) = Y )
     => ( ( ( Xa2 = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V: nat] : Xa2 = some_nat(V)
           => ( ( Xb = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A3: nat] :
                ( ( Xa2 = some_nat(A3) )
               => ! [B3: nat] :
                    ( ( Xb = some_nat(B3) )
                   => ( Y != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(X,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1732_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(plus_plus_real(Y),E2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_epsilon
tff(fact_1733_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),E2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_epsilon
tff(fact_1734_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(plus_plus_real(A),B)),zero_zero_real)) ) ) ).

% add_neg_nonpos
tff(fact_1735_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).

% add_neg_nonpos
tff(fact_1736_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_1737_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_1738_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(plus_plus_real(A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1739_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1740_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_1741_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_1742_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(plus_plus_real(A),B)),zero_zero_real)) ) ) ).

% add_nonpos_neg
tff(fact_1743_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).

% add_nonpos_neg
tff(fact_1744_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_1745_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_1746_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(plus_plus_real(A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1747_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1748_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_1749_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_1750_add__strict__increasing,axiom,
    ! [A: real,B: real,C2: 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),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(plus_plus_real(A),C2))) ) ) ).

% add_strict_increasing
tff(fact_1751_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2))) ) ) ).

% add_strict_increasing
tff(fact_1752_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => 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),C2))) ) ) ).

% add_strict_increasing
tff(fact_1753_add__strict__increasing,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => 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),C2))) ) ) ).

% add_strict_increasing
tff(fact_1754_add__strict__increasing2,axiom,
    ! [A: real,B: real,C2: 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),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(plus_plus_real(A),C2))) ) ) ).

% add_strict_increasing2
tff(fact_1755_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2))) ) ) ).

% add_strict_increasing2
tff(fact_1756_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => 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),C2))) ) ) ).

% add_strict_increasing2
tff(fact_1757_add__strict__increasing2,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => 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),C2))) ) ) ).

% add_strict_increasing2
tff(fact_1758_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(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_1759_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_1760_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_1761_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_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_1762_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat)) ).

% not_sum_squares_lt_zero
tff(fact_1763_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_1764_discrete,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_le3102999989581377725nteger,plus_p5714425477246183910nteger(A,one_one_Code_integer)),B)) ) ).

% discrete
tff(fact_1765_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_1766_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_1767_zero__less__two,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,one_one_Code_integer))) ).

% zero_less_two
tff(fact_1768_zero__less__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(one_one_real),one_one_real))) ).

% zero_less_two
tff(fact_1769_zero__less__two,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ).

% zero_less_two
tff(fact_1770_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_1771_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_1772_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1773_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1774_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1775_convex__bound__le,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),A))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V2))
           => ( ( plus_p5714425477246183910nteger(U,V2) = one_one_Code_integer )
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,U),X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V2),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1776_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V2: 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),V2))
           => ( ( aa_real_real(plus_plus_real(U),V2) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V2),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1777_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V2: 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),V2))
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V2) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V2),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1778_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V2: 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),V2))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V2) = 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,V2),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1779_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,Uw: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy,Uz2)
               => ( Y != some_nat(Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_1780_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,Uw: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy,Uz2)
               => ( Y != some_nat(Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_1781_vebt__mint_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A3: bool,B3: bool] : X != vEBT_Leaf(A3,B3)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw)
       => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy,Uz2) ) ) ).

% vebt_mint.cases
tff(fact_1782_convex__bound__lt,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),A))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V2))
           => ( ( plus_p5714425477246183910nteger(U,V2) = one_one_Code_integer )
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,U),X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V2),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1783_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V2: 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),V2))
           => ( ( aa_real_real(plus_plus_real(U),V2) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V2),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1784_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V2: 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),V2))
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V2) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V2),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1785_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V2: 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),V2))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V2) = 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,V2),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1786_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va3: 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,Va3,Vb)),X))
    <=> ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_1787_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X3)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw),X3)
       => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz2),X3)
         => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X3)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: 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),X3) ) ) ) ) ).

% vebt_member.cases
tff(fact_1788_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_1789_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_1790_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: 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)),N))
     => ( 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_1791_vebt__pred_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),zero_zero_nat)
     => ( ! [A3: bool,Uw: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,Uw),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)))
         => ( ! [Uy: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz2,Va2),Vb2)
           => ( ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve),Vf)
             => ( ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Vj)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: 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),X3) ) ) ) ) ) ) ).

% vebt_pred.cases
tff(fact_1792_vebt__succ_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,B3: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,B3),zero_zero_nat)
     => ( ! [Uv2: bool,Uw: bool,N3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uv2,Uw),aa_nat_nat(suc,N3))
       => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy,Uz2),Va2)
         => ( ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2,Vd),Ve)
           => ( ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Vi)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: 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),X3) ) ) ) ) ) ).

% vebt_succ.cases
tff(fact_1793_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,N3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),aa_nat_nat(suc,aa_nat_nat(suc,N3)))
         => ( ! [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,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst,Smry),X3)
             => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X3: 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),Tr,Sm),X3)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: 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),X3) ) ) ) ) ) ) ).

% vebt_delete.cases
tff(fact_1794_vebt__insert_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X3)
     => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info,zero_zero_nat,Ts,S),X3)
       => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S),X3)
         => ( ! [V: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary2),X3)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: 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),X3) ) ) ) ) ).

% vebt_insert.cases
tff(fact_1795_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( 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_1796_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( 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_1797_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( 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_1798_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( 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_1799_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_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_1800_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_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_1801_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_1802_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: bool,Uw2: bool,N: nat] : vEBT_vebt_succ(vEBT_Leaf(Uv,Uw2),aa_nat_nat(suc,N)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_1803_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_1804_vebt__succ_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc,Vd2),Ve2) = none_nat ).

% vebt_succ.simps(4)
tff(fact_1805_vebt__pred_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2),Vf2) = none_nat ).

% vebt_pred.simps(5)
tff(fact_1806_vebt__pred_Osimps_I6_J,axiom,
    ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Vj2) = none_nat ).

% vebt_pred.simps(6)
tff(fact_1807_vebt__succ_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Vi2) = none_nat ).

% vebt_succ.simps(5)
tff(fact_1808_vebt__pred_Osimps_I2_J,axiom,
    ! [A: bool,Uw2: bool] :
      ( ( pp(A)
       => ( vEBT_vebt_pred(vEBT_Leaf(A,Uw2),aa_nat_nat(suc,zero_zero_nat)) = some_nat(zero_zero_nat) ) )
      & ( ~ pp(A)
       => ( vEBT_vebt_pred(vEBT_Leaf(A,Uw2),aa_nat_nat(suc,zero_zero_nat)) = none_nat ) ) ) ).

% vebt_pred.simps(2)
tff(fact_1809_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_1810_vebt__pred_Osimps_I3_J,axiom,
    ! [B: bool,A: bool,Va3: nat] :
      ( ( pp(B)
       => ( vEBT_vebt_pred(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va3))) = some_nat(one_one_nat) ) )
      & ( ~ pp(B)
       => ( ( pp(A)
           => ( vEBT_vebt_pred(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va3))) = some_nat(zero_zero_nat) ) )
          & ( ~ pp(A)
           => ( vEBT_vebt_pred(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va3))) = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
tff(fact_1811_vebt__delete_Osimps_I3_J,axiom,
    ! [A: bool,B: bool,N: nat] : vEBT_vebt_delete(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,N))) = vEBT_Leaf(A,B) ).

% vebt_delete.simps(3)
tff(fact_1812_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_1813_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))
        & ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),X))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y)) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_1814_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))
        & ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X4)) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_1815_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_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_1816_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_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_1817_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_1818_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(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_1819_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1820_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_1821_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_1822_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst2,Smry2),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst2,Smry2) ).

% vebt_delete.simps(5)
tff(fact_1823_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr2: list_VEBT_VEBT,Sm2: 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),Tr2,Sm2),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) ).

% vebt_delete.simps(6)
tff(fact_1824_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_1825_delete__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T,X)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,vEBT_set_vebt(T)),insert_nat(X,bot_bot_set_nat)) ) ) ).

% delete_correct
tff(fact_1826_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(plus_plus_real(A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_1827_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_1828_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_1829_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_1830_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T,X)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,vEBT_VEBT_set_vebt(T)),insert_nat(X,bot_bot_set_nat)) ) ) ).

% delete_correct'
tff(fact_1831_obtain__set__succ,axiom,
    ! [X: nat,Z2: nat,A4: set_nat,B5: 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(A4),Z2))
       => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
         => ( ( A4 = B5 )
           => ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(A4),X),X_12)) ) ) ) ) ).

% obtain_set_succ
tff(fact_1832_obtain__set__pred,axiom,
    ! [Z2: nat,X: nat,A4: 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(A4),Z2))
       => ( pp(aa_set_nat_bool(finite_finite_nat,A4))
         => ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(A4),X),X_12)) ) ) ) ).

% obtain_set_pred
tff(fact_1833_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_1834_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( pp(aa_set_real_bool(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_1835_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_1836_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_1837_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_1838_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),aa_list_nat_nat(size_size_list_nat,Xs))) ) ).

% length_pos_if_in_set
tff(fact_1839_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_1840_option_Osize__gen_I2_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat,X2: product_prod_nat_nat] : size_o8335143837870341156at_nat(X,some_P7363390416028606310at_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1841_option_Osize__gen_I2_J,axiom,
    ! [X: fun_nat_nat,X2: nat] : size_option_nat(X,some_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1842_option_Osize__gen_I2_J,axiom,
    ! [X: fun_num_nat,X2: num] : size_option_num(X,some_num(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1843_add__scale__eq__noteq,axiom,
    ! [R: complex,A: complex,B: complex,C2: complex,D: complex] :
      ( ( R != zero_zero_complex )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R),C2)) != aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1844_add__scale__eq__noteq,axiom,
    ! [R: real,A: real,B: real,C2: real,D: real] :
      ( ( R != zero_zero_real )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_real_real(plus_plus_real(A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),C2)) != aa_real_real(plus_plus_real(B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1845_add__scale__eq__noteq,axiom,
    ! [R: rat,A: rat,B: rat,C2: rat,D: rat] :
      ( ( R != zero_zero_rat )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),C2)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1846_add__scale__eq__noteq,axiom,
    ! [R: nat,A: nat,B: nat,C2: nat,D: nat] :
      ( ( R != zero_zero_nat )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),C2)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1847_add__scale__eq__noteq,axiom,
    ! [R: int,A: int,B: int,C2: int,D: int] :
      ( ( R != zero_zero_int )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R),C2)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1848_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => pp(aa_set_nat_bool(finite_finite_nat,vEBT_VEBT_set_vebt(T))) ) ).

% set_vebt_finite
tff(fact_1849_pred__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(Xs),A),X_12))
     => ( pp(aa_set_nat_bool(finite_finite_nat,Xs))
       => ~ ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),A)) ) ) ) ).

% pred_none_empty
tff(fact_1850_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(Xs),A),X_12))
     => ( pp(aa_set_nat_bool(finite_finite_nat,Xs))
       => ~ ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),X5)) ) ) ) ).

% succ_none_empty
tff(fact_1851_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% diff_self
tff(fact_1852_diff__self,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),A) = zero_zero_real ).

% diff_self
tff(fact_1853_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),A) = zero_zero_rat ).

% diff_self
tff(fact_1854_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_1855_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_1856_diff__0__right,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_0_right
tff(fact_1857_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_1858_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_1859_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_1860_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_zero
tff(fact_1861_diff__zero,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_zero
tff(fact_1862_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),zero_zero_rat) = A ).

% diff_zero
tff(fact_1863_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_1864_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_1865_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1866_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),A) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1867_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1868_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_1869_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_1870_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_1871_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_1872_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_1873_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_1874_add__diff__cancel__right,axiom,
    ! [A: real,C2: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C2)),aa_real_real(plus_plus_real(B),C2)) = aa_real_real(minus_minus_real(A),B) ).

% add_diff_cancel_right
tff(fact_1875_add__diff__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ).

% add_diff_cancel_right
tff(fact_1876_add__diff__cancel__right,axiom,
    ! [A: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_1877_add__diff__cancel__right,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_right
tff(fact_1878_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_1879_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_1880_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_1881_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_1882_add__diff__cancel__left,axiom,
    ! [C2: real,A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(C2),A)),aa_real_real(plus_plus_real(C2),B)) = aa_real_real(minus_minus_real(A),B) ).

% add_diff_cancel_left
tff(fact_1883_add__diff__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ).

% add_diff_cancel_left
tff(fact_1884_add__diff__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_1885_add__diff__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_left
tff(fact_1886_diff__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),B) = A ).

% diff_add_cancel
tff(fact_1887_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),B) = A ).

% diff_add_cancel
tff(fact_1888_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_1889_add__diff__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),B) = A ).

% add_diff_cancel
tff(fact_1890_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ).

% add_diff_cancel
tff(fact_1891_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_1892_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(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_1893_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(aa_rat_fun_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_1894_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_1895_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(minus_minus_real(A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_1896_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(aa_rat_fun_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_1897_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_1898_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(plus_plus_real(aa_real_real(minus_minus_real(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_1899_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_1900_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_1901_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_1902_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(plus_plus_real(B),aa_real_real(minus_minus_real(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_1903_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(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_1904_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_1905_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_1906_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_1907_diff__numeral__special_I9_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% diff_numeral_special(9)
tff(fact_1908_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_1909_diff__numeral__special_I9_J,axiom,
    aa_real_real(minus_minus_real(one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_1910_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_1911_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_1912_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) != size_s6755466524823107622T_VEBT(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1913_neq__if__length__neq,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( size_size_list_bool(Xs) != size_size_list_bool(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1914_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) != aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1915_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) != size_size_list_int(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1916_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = N ).

% Ex_list_of_length
tff(fact_1917_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_bool] : size_size_list_bool(Xs2) = N ).

% Ex_list_of_length
tff(fact_1918_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] : aa_list_nat_nat(size_size_list_nat,Xs2) = N ).

% Ex_list_of_length
tff(fact_1919_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = N ).

% Ex_list_of_length
tff(fact_1920_diff__right__commute,axiom,
    ! [A: rat,C2: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C2)),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C2) ).

% diff_right_commute
tff(fact_1921_diff__right__commute,axiom,
    ! [A: nat,C2: 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),C2)),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)),C2) ).

% diff_right_commute
tff(fact_1922_diff__right__commute,axiom,
    ! [A: int,C2: 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),C2)),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)),C2) ).

% diff_right_commute
tff(fact_1923_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),D) )
     => ( ( A = B )
      <=> ( C2 = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1924_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C2: 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,C2),D) )
     => ( ( A = B )
      <=> ( C2 = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1925_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D))) ) ) ).

% diff_mono
tff(fact_1926_diff__mono,axiom,
    ! [A: int,B: int,D: int,C2: 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),C2))
       => 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),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))) ) ) ).

% diff_mono
tff(fact_1927_diff__left__mono,axiom,
    ! [B: rat,A: rat,C2: 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(aa_rat_fun_rat_rat(minus_minus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),B))) ) ).

% diff_left_mono
tff(fact_1928_diff__left__mono,axiom,
    ! [B: int,A: int,C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B))) ) ).

% diff_left_mono
tff(fact_1929_diff__right__mono,axiom,
    ! [A: rat,B: rat,C2: 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(aa_rat_fun_rat_rat(minus_minus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C2))) ) ).

% diff_right_mono
tff(fact_1930_diff__right__mono,axiom,
    ! [A: int,B: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2))) ) ).

% diff_right_mono
tff(fact_1931_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),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,C2),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1932_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C2: 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,C2),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,C2),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1933_eq__iff__diff__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = B )
    <=> ( aa_complex_complex(minus_minus_complex(A),B) = zero_zero_complex ) ) ).

% eq_iff_diff_eq_0
tff(fact_1934_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( aa_real_real(minus_minus_real(A),B) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_1935_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_1936_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_1937_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C2: 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(minus_minus_real(A),C2)),aa_real_real(minus_minus_real(B),C2))) ) ).

% diff_strict_right_mono
tff(fact_1938_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: 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(aa_rat_fun_rat_rat(minus_minus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C2))) ) ).

% diff_strict_right_mono
tff(fact_1939_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2))) ) ).

% diff_strict_right_mono
tff(fact_1940_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C2: 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(minus_minus_real(C2),A)),aa_real_real(minus_minus_real(C2),B))) ) ).

% diff_strict_left_mono
tff(fact_1941_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C2: 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(aa_rat_fun_rat_rat(minus_minus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),B))) ) ).

% diff_strict_left_mono
tff(fact_1942_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B))) ) ).

% diff_strict_left_mono
tff(fact_1943_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C2),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),D)) ) ) ).

% diff_eq_diff_less
tff(fact_1944_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),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,C2),D)) ) ) ).

% diff_eq_diff_less
tff(fact_1945_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C2: 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,C2),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,C2),D)) ) ) ).

% diff_eq_diff_less
tff(fact_1946_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),C2)),aa_real_real(minus_minus_real(B),D))) ) ) ).

% diff_strict_mono
tff(fact_1947_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D))) ) ) ).

% diff_strict_mono
tff(fact_1948_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C2: 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),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))) ) ) ).

% diff_strict_mono
tff(fact_1949_left__diff__distrib,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),B)),C2) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% left_diff_distrib
tff(fact_1950_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ).

% left_diff_distrib
tff(fact_1951_left__diff__distrib,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% left_diff_distrib
tff(fact_1952_right__diff__distrib,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(minus_minus_real(B),C2)) = aa_real_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),C2)) ).

% right_diff_distrib
tff(fact_1953_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)) ).

% right_diff_distrib
tff(fact_1954_right__diff__distrib,axiom,
    ! [A: int,B: int,C2: 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),C2)) = 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),C2)) ).

% right_diff_distrib
tff(fact_1955_left__diff__distrib_H,axiom,
    ! [B: real,C2: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),C2)),A) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)) ).

% left_diff_distrib'
tff(fact_1956_left__diff__distrib_H,axiom,
    ! [B: rat,C2: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C2)),A) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)) ).

% left_diff_distrib'
tff(fact_1957_left__diff__distrib_H,axiom,
    ! [B: nat,C2: 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),C2)),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,C2),A)) ).

% left_diff_distrib'
tff(fact_1958_left__diff__distrib_H,axiom,
    ! [B: int,C2: 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),C2)),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,C2),A)) ).

% left_diff_distrib'
tff(fact_1959_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(minus_minus_real(B),C2)) = aa_real_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),C2)) ).

% right_diff_distrib'
tff(fact_1960_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)) ).

% right_diff_distrib'
tff(fact_1961_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C2: 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),C2)) = 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),C2)) ).

% right_diff_distrib'
tff(fact_1962_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C2: 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),C2)) = 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),C2)) ).

% right_diff_distrib'
tff(fact_1963_diff__diff__eq,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B)),C2) = aa_real_real(minus_minus_real(A),aa_real_real(plus_plus_real(B),C2)) ).

% diff_diff_eq
tff(fact_1964_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) ).

% diff_diff_eq
tff(fact_1965_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C2: 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)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% diff_diff_eq
tff(fact_1966_diff__diff__eq,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% diff_diff_eq
tff(fact_1967_add__implies__diff,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( aa_real_real(plus_plus_real(C2),B) = A )
     => ( C2 = aa_real_real(minus_minus_real(A),B) ) ) ).

% add_implies_diff
tff(fact_1968_add__implies__diff,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B) = A )
     => ( C2 = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ) ) ).

% add_implies_diff
tff(fact_1969_add__implies__diff,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B) = A )
     => ( C2 = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_1970_add__implies__diff,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B) = A )
     => ( C2 = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_1971_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(minus_minus_real(A),aa_real_real(plus_plus_real(B),C2)) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C2)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_1972_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C2)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_1973_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C2: 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),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_1974_diff__add__eq,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),C2) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C2)),B) ).

% diff_add_eq
tff(fact_1975_diff__add__eq,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),B) ).

% diff_add_eq
tff(fact_1976_diff__add__eq,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ).

% diff_add_eq
tff(fact_1977_diff__diff__eq2,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(minus_minus_real(A),aa_real_real(minus_minus_real(B),C2)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C2)),B) ).

% diff_diff_eq2
tff(fact_1978_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),B) ).

% diff_diff_eq2
tff(fact_1979_diff__diff__eq2,axiom,
    ! [A: int,B: int,C2: 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),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ).

% diff_diff_eq2
tff(fact_1980_add__diff__eq,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(plus_plus_real(A),aa_real_real(minus_minus_real(B),C2)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),C2) ).

% add_diff_eq
tff(fact_1981_add__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C2) ).

% add_diff_eq
tff(fact_1982_add__diff__eq,axiom,
    ! [A: int,B: int,C2: 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),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ).

% add_diff_eq
tff(fact_1983_eq__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( A = aa_real_real(minus_minus_real(C2),B) )
    <=> ( aa_real_real(plus_plus_real(A),B) = C2 ) ) ).

% eq_diff_eq
tff(fact_1984_eq__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = C2 ) ) ).

% eq_diff_eq
tff(fact_1985_eq__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C2 ) ) ).

% eq_diff_eq
tff(fact_1986_diff__eq__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = C2 )
    <=> ( A = aa_real_real(plus_plus_real(C2),B) ) ) ).

% diff_eq_eq
tff(fact_1987_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = C2 )
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B) ) ) ).

% diff_eq_eq
tff(fact_1988_diff__eq__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C2 )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B) ) ) ).

% diff_eq_eq
tff(fact_1989_group__cancel_Osub1,axiom,
    ! [A4: real,K2: real,A: real,B: real] :
      ( ( A4 = aa_real_real(plus_plus_real(K2),A) )
     => ( aa_real_real(minus_minus_real(A4),B) = aa_real_real(plus_plus_real(K2),aa_real_real(minus_minus_real(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_1990_group__cancel_Osub1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A4),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_1991_group__cancel_Osub1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A4),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_1992_finite__nat__set__iff__bounded,axiom,
    ! [N5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
    <=> ? [M6: 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),M6)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_1993_bounded__nat__set__is__finite,axiom,
    ! [N5: set_nat,N: 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),N)) )
     => pp(aa_set_nat_bool(finite_finite_nat,N5)) ) ).

% bounded_nat_set_is_finite
tff(fact_1994_finite__nat__set__iff__bounded__le,axiom,
    ! [N5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
    <=> ? [M6: nat] :
        ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),N5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),M6)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_1995_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(minus_minus_real(A),B)),zero_zero_real)) ) ).

% le_iff_diff_le_0
tff(fact_1996_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(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat)) ) ).

% le_iff_diff_le_0
tff(fact_1997_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_1998_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(minus_minus_real(A),B)),zero_zero_real)) ) ).

% less_iff_diff_less_0
tff(fact_1999_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(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat)) ) ).

% less_iff_diff_less_0
tff(fact_2000_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_2001_add__le__add__imp__diff__le,axiom,
    ! [I2: real,K2: real,N: real,J2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(I2),K2)),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(plus_plus_real(J2),K2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(I2),K2)),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(plus_plus_real(J2),K2)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(N),K2)),J2)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2002_add__le__add__imp__diff__le,axiom,
    ! [I2: rat,K2: rat,N: rat,J2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),K2)))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),K2)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,N),K2)),J2)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2003_add__le__add__imp__diff__le,axiom,
    ! [I2: nat,K2: nat,N: nat,J2: 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,I2),K2)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)),J2)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2004_add__le__add__imp__diff__le,axiom,
    ! [I2: int,K2: int,N: int,J2: 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,I2),K2)),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),K2)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),K2)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K2)),J2)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2005_add__le__imp__le__diff,axiom,
    ! [I2: real,K2: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(I2),K2)),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I2),aa_real_real(minus_minus_real(N),K2))) ) ).

% add_le_imp_le_diff
tff(fact_2006_add__le__imp__le__diff,axiom,
    ! [I2: rat,K2: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,N),K2))) ) ).

% add_le_imp_le_diff
tff(fact_2007_add__le__imp__le__diff,axiom,
    ! [I2: nat,K2: nat,N: 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,I2),K2)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2))) ) ).

% add_le_imp_le_diff
tff(fact_2008_add__le__imp__le__diff,axiom,
    ! [I2: int,K2: int,N: 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,I2),K2)),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K2))) ) ).

% add_le_imp_le_diff
tff(fact_2009_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C2: 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) = C2 )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_2010_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_2011_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C2: 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,C2),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,C2),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_2012_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C2: 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),C2)),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)),C2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_2013_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C2: 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)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_2014_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C2: 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,C2),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_2015_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C2: 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,C2),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,C2),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_2016_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C2: 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,C2),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,C2),A)),B)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_2017_le__add__diff,axiom,
    ! [A: nat,B: nat,C2: 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,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)),A))) ) ).

% le_add_diff
tff(fact_2018_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_2019_le__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(minus_minus_real(C2),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(A),B)),C2)) ) ).

% le_diff_eq
tff(fact_2020_le__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C2)) ) ).

% le_diff_eq
tff(fact_2021_le__diff__eq,axiom,
    ! [A: int,C2: 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,C2),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)),C2)) ) ).

% le_diff_eq
tff(fact_2022_diff__le__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(A),B)),C2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(plus_plus_real(C2),B))) ) ).

% diff_le_eq
tff(fact_2023_diff__le__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B))) ) ).

% diff_le_eq
tff(fact_2024_diff__le__eq,axiom,
    ! [A: int,B: int,C2: 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)),C2))
    <=> 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,C2),B))) ) ).

% diff_le_eq
tff(fact_2025_less__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(minus_minus_real(C2),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(A),B)),C2)) ) ).

% less_diff_eq
tff(fact_2026_less__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C2)) ) ).

% less_diff_eq
tff(fact_2027_less__diff__eq,axiom,
    ! [A: int,C2: 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,C2),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)),C2)) ) ).

% less_diff_eq
tff(fact_2028_diff__less__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),B)),C2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(plus_plus_real(C2),B))) ) ).

% diff_less_eq
tff(fact_2029_diff__less__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B))) ) ).

% diff_less_eq
tff(fact_2030_diff__less__eq,axiom,
    ! [A: int,B: int,C2: 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)),C2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))) ) ).

% diff_less_eq
tff(fact_2031_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(plus_plus_real(B),aa_real_real(minus_minus_real(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_2032_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(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_2033_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_2034_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_2035_square__diff__square__factored,axiom,
    ! [X: real,Y: real] : aa_real_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(plus_plus_real(X),Y)),aa_real_real(minus_minus_real(X),Y)) ).

% square_diff_square_factored
tff(fact_2036_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)) ).

% square_diff_square_factored
tff(fact_2037_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_2038_eq__add__iff2,axiom,
    ! [A: real,E: real,C2: real,B: real,D: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C2) = aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D) )
    <=> ( C2 = aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_2039_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D) )
    <=> ( C2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_2040_eq__add__iff2,axiom,
    ! [A: int,E: int,C2: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
    <=> ( C2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_2041_eq__add__iff1,axiom,
    ! [A: real,E: real,C2: real,B: real,D: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C2) = aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D) )
    <=> ( aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),B)),E)),C2) = D ) ) ).

% eq_add_iff1
tff(fact_2042_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E)),C2) = D ) ) ).

% eq_add_iff1
tff(fact_2043_eq__add__iff1,axiom,
    ! [A: int,E: int,C2: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C2) = D ) ) ).

% eq_add_iff1
tff(fact_2044_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C2: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C2)),aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),A)),E)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_2045_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_2046_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C2: 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),E)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_2047_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C2: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C2)),aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),B)),E)),C2)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_2048_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E)),C2)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_2049_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C2: 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),E)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D)))
    <=> 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)),E)),C2)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_2050_less__add__iff1,axiom,
    ! [A: real,E: real,C2: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C2)),aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),B)),E)),C2)),D)) ) ).

% less_add_iff1
tff(fact_2051_less__add__iff1,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E)),C2)),D)) ) ).

% less_add_iff1
tff(fact_2052_less__add__iff1,axiom,
    ! [A: int,E: int,C2: 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),E)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D)))
    <=> 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)),E)),C2)),D)) ) ).

% less_add_iff1
tff(fact_2053_less__add__iff2,axiom,
    ! [A: real,E: real,C2: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C2)),aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),A)),E)),D))) ) ).

% less_add_iff2
tff(fact_2054_less__add__iff2,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E)),D))) ) ).

% less_add_iff2
tff(fact_2055_less__add__iff2,axiom,
    ! [A: int,E: int,C2: 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),E)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D))) ) ).

% less_add_iff2
tff(fact_2056_square__diff__one__factored,axiom,
    ! [X: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(X,one_one_Code_integer)),minus_8373710615458151222nteger(X,one_one_Code_integer)) ).

% square_diff_one_factored
tff(fact_2057_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_2058_square__diff__one__factored,axiom,
    ! [X: real] : aa_real_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(plus_plus_real(X),one_one_real)),aa_real_real(minus_minus_real(X),one_one_real)) ).

% square_diff_one_factored
tff(fact_2059_square__diff__one__factored,axiom,
    ! [X: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),one_one_rat)) ).

% square_diff_one_factored
tff(fact_2060_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_2061_finite__ranking__induct,axiom,
    ! [S3: set_complex,P2: fun_set_complex_bool,F2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( pp(aa_set_complex_bool(P2,bot_bot_set_complex))
       => ( ! [X3: complex,S4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S4))
             => ( ! [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),S4))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F2,Y4)),aa_complex_rat(F2,X3))) )
               => ( pp(aa_set_complex_bool(P2,S4))
                 => pp(aa_set_complex_bool(P2,insert_complex(X3,S4))) ) ) )
         => pp(aa_set_complex_bool(P2,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_2062_finite__ranking__induct,axiom,
    ! [S3: set_nat,P2: fun_set_nat_bool,F2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( pp(aa_set_nat_bool(P2,bot_bot_set_nat))
       => ( ! [X3: nat,S4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,S4))
             => ( ! [Y4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y4),S4))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,Y4)),aa_nat_rat(F2,X3))) )
               => ( pp(aa_set_nat_bool(P2,S4))
                 => pp(aa_set_nat_bool(P2,insert_nat(X3,S4))) ) ) )
         => pp(aa_set_nat_bool(P2,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_2063_finite__ranking__induct,axiom,
    ! [S3: set_int,P2: fun_set_int_bool,F2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( pp(aa_set_int_bool(P2,bot_bot_set_int))
       => ( ! [X3: int,S4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,S4))
             => ( ! [Y4: int] :
                    ( pp(aa_set_int_bool(member_int(Y4),S4))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F2,Y4)),aa_int_rat(F2,X3))) )
               => ( pp(aa_set_int_bool(P2,S4))
                 => pp(aa_set_int_bool(P2,insert_int(X3,S4))) ) ) )
         => pp(aa_set_int_bool(P2,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_2064_finite__ranking__induct,axiom,
    ! [S3: set_real,P2: fun_set_real_bool,F2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( pp(aa_set_real_bool(P2,bot_bot_set_real))
       => ( ! [X3: real,S4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S4))
             => ( ! [Y4: real] :
                    ( pp(aa_set_real_bool(member_real(Y4),S4))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F2,Y4)),aa_real_rat(F2,X3))) )
               => ( pp(aa_set_real_bool(P2,S4))
                 => pp(aa_set_real_bool(P2,insert_real(X3,S4))) ) ) )
         => pp(aa_set_real_bool(P2,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_2065_finite__ranking__induct,axiom,
    ! [S3: set_complex,P2: fun_set_complex_bool,F2: fun_complex_num] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( pp(aa_set_complex_bool(P2,bot_bot_set_complex))
       => ( ! [X3: complex,S4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S4))
             => ( ! [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),S4))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_complex_num(F2,Y4)),aa_complex_num(F2,X3))) )
               => ( pp(aa_set_complex_bool(P2,S4))
                 => pp(aa_set_complex_bool(P2,insert_complex(X3,S4))) ) ) )
         => pp(aa_set_complex_bool(P2,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_2066_finite__ranking__induct,axiom,
    ! [S3: set_nat,P2: fun_set_nat_bool,F2: fun_nat_num] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( pp(aa_set_nat_bool(P2,bot_bot_set_nat))
       => ( ! [X3: nat,S4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,S4))
             => ( ! [Y4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y4),S4))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F2,Y4)),aa_nat_num(F2,X3))) )
               => ( pp(aa_set_nat_bool(P2,S4))
                 => pp(aa_set_nat_bool(P2,insert_nat(X3,S4))) ) ) )
         => pp(aa_set_nat_bool(P2,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_2067_finite__ranking__induct,axiom,
    ! [S3: set_int,P2: fun_set_int_bool,F2: fun_int_num] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( pp(aa_set_int_bool(P2,bot_bot_set_int))
       => ( ! [X3: int,S4: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,S4))
             => ( ! [Y4: int] :
                    ( pp(aa_set_int_bool(member_int(Y4),S4))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F2,Y4)),aa_int_num(F2,X3))) )
               => ( pp(aa_set_int_bool(P2,S4))
                 => pp(aa_set_int_bool(P2,insert_int(X3,S4))) ) ) )
         => pp(aa_set_int_bool(P2,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_2068_finite__ranking__induct,axiom,
    ! [S3: set_real,P2: fun_set_real_bool,F2: fun_real_num] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( pp(aa_set_real_bool(P2,bot_bot_set_real))
       => ( ! [X3: real,S4: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S4))
             => ( ! [Y4: real] :
                    ( pp(aa_set_real_bool(member_real(Y4),S4))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_real_num(F2,Y4)),aa_real_num(F2,X3))) )
               => ( pp(aa_set_real_bool(P2,S4))
                 => pp(aa_set_real_bool(P2,insert_real(X3,S4))) ) ) )
         => pp(aa_set_real_bool(P2,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_2069_finite__ranking__induct,axiom,
    ! [S3: set_complex,P2: fun_set_complex_bool,F2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( pp(aa_set_complex_bool(P2,bot_bot_set_complex))
       => ( ! [X3: complex,S4: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S4))
             => ( ! [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),S4))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F2,Y4)),aa_complex_nat(F2,X3))) )
               => ( pp(aa_set_complex_bool(P2,S4))
                 => pp(aa_set_complex_bool(P2,insert_complex(X3,S4))) ) ) )
         => pp(aa_set_complex_bool(P2,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_2070_finite__ranking__induct,axiom,
    ! [S3: set_nat,P2: fun_set_nat_bool,F2: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( pp(aa_set_nat_bool(P2,bot_bot_set_nat))
       => ( ! [X3: nat,S4: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,S4))
             => ( ! [Y4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y4),S4))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F2,Y4)),aa_nat_nat(F2,X3))) )
               => ( pp(aa_set_nat_bool(P2,S4))
                 => pp(aa_set_nat_bool(P2,insert_nat(X3,S4))) ) ) )
         => pp(aa_set_nat_bool(P2,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_2071_length__induct,axiom,
    ! [P2: fun_li6006399982103069984T_bool,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(Ys2)),size_s6755466524823107622T_VEBT(Xs2)))
             => pp(aa_lis2822252094023462681T_bool(P2,Ys2)) )
         => pp(aa_lis2822252094023462681T_bool(P2,Xs2)) )
     => pp(aa_lis2822252094023462681T_bool(P2,Xs)) ) ).

% length_induct
tff(fact_2072_length__induct,axiom,
    ! [P2: fun_list_bool_bool,Xs: list_bool] :
      ( ! [Xs2: list_bool] :
          ( ! [Ys2: list_bool] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(Ys2)),size_size_list_bool(Xs2)))
             => pp(aa_list_bool_bool(P2,Ys2)) )
         => pp(aa_list_bool_bool(P2,Xs2)) )
     => pp(aa_list_bool_bool(P2,Xs)) ) ).

% length_induct
tff(fact_2073_length__induct,axiom,
    ! [P2: fun_list_nat_bool,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_nat_nat(size_size_list_nat,Ys2)),aa_list_nat_nat(size_size_list_nat,Xs2)))
             => pp(aa_list_nat_bool(P2,Ys2)) )
         => pp(aa_list_nat_bool(P2,Xs2)) )
     => pp(aa_list_nat_bool(P2,Xs)) ) ).

% length_induct
tff(fact_2074_length__induct,axiom,
    ! [P2: fun_list_int_bool,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs2)))
             => pp(aa_list_int_bool(P2,Ys2)) )
         => pp(aa_list_int_bool(P2,Xs2)) )
     => pp(aa_list_int_bool(P2,Xs)) ) ).

% length_induct
tff(fact_2075_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( A = zero_zero_complex ) ) ).

% add_0_iff
tff(fact_2076_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B = aa_real_real(plus_plus_real(B),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_2077_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_2078_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_2079_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_2080_crossproduct__eq,axiom,
    ! [W2: real,Y: real,X: real,Z2: real] :
      ( ( aa_real_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(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_2081_crossproduct__eq,axiom,
    ! [W2: rat,Y: rat,X: rat,Z2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Z2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
tff(fact_2082_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_2083_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_2084_crossproduct__noteq,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ( A != B )
        & ( C2 != D ) )
    <=> ( aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) != aa_real_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),C2)) ) ) ).

% crossproduct_noteq
tff(fact_2085_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ( A != B )
        & ( C2 != D ) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),D)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ) ) ).

% crossproduct_noteq
tff(fact_2086_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ( A != B )
        & ( C2 != D ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),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),C2)) ) ) ).

% crossproduct_noteq
tff(fact_2087_crossproduct__noteq,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ( A != B )
        & ( C2 != D ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),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),C2)) ) ) ).

% crossproduct_noteq
tff(fact_2088_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_2089_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_2090_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_2091_finite__has__minimal,axiom,
    ! [A4: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( A4 != bot_bot_set_real )
       => ? [X3: real] :
            ( pp(aa_set_real_bool(member_real(X3),A4))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X3))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_2092_finite__has__minimal,axiom,
    ! [A4: set_set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( ( A4 != bot_bot_set_set_nat )
       => ? [X3: set_nat] :
            ( member_set_nat(X3,A4)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A4)
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Xa),X3))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_2093_finite__has__minimal,axiom,
    ! [A4: set_rat] :
      ( finite_finite_rat(A4)
     => ( ( A4 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( member_rat(X3,A4)
            & ! [Xa: rat] :
                ( member_rat(Xa,A4)
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa),X3))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_2094_finite__has__minimal,axiom,
    ! [A4: set_num] :
      ( finite_finite_num(A4)
     => ( ( A4 != bot_bot_set_num )
       => ? [X3: num] :
            ( member_num(X3,A4)
            & ! [Xa: num] :
                ( member_num(Xa,A4)
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa),X3))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_2095_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( A4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A4))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A4))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X3))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_2096_finite__has__minimal,axiom,
    ! [A4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( A4 != bot_bot_set_int )
       => ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A4))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A4))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X3))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_2097_finite__has__maximal,axiom,
    ! [A4: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( A4 != bot_bot_set_real )
       => ? [X3: real] :
            ( pp(aa_set_real_bool(member_real(X3),A4))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Xa))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_2098_finite__has__maximal,axiom,
    ! [A4: set_set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( ( A4 != bot_bot_set_set_nat )
       => ? [X3: set_nat] :
            ( member_set_nat(X3,A4)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A4)
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X3),Xa))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_2099_finite__has__maximal,axiom,
    ! [A4: set_rat] :
      ( finite_finite_rat(A4)
     => ( ( A4 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( member_rat(X3,A4)
            & ! [Xa: rat] :
                ( member_rat(Xa,A4)
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Xa))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_2100_finite__has__maximal,axiom,
    ! [A4: set_num] :
      ( finite_finite_num(A4)
     => ( ( A4 != bot_bot_set_num )
       => ? [X3: num] :
            ( member_num(X3,A4)
            & ! [Xa: num] :
                ( member_num(Xa,A4)
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Xa))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_2101_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( A4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A4))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A4))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Xa))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_2102_finite__has__maximal,axiom,
    ! [A4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( A4 != bot_bot_set_int )
       => ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A4))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A4))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Xa))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_2103_mult__diff__mult,axiom,
    ! [X: real,Y: real,A: real,B: real] : aa_real_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(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(minus_minus_real(Y),B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(X),A)),B)) ).

% mult_diff_mult
tff(fact_2104_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Y),B))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),A)),B)) ).

% mult_diff_mult
tff(fact_2105_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_2106_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_2107_diff__shunt__var,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( minus_minus_set_real(X,Y) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(X),Y)) ) ).

% diff_shunt_var
tff(fact_2108_diff__shunt__var,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,X),Y) = bot_bo2099793752762293965at_nat )
    <=> ord_le3146513528884898305at_nat(X,Y) ) ).

% diff_shunt_var
tff(fact_2109_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_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_2110_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ( vEBT_vebt_mint(T) = none_nat ) ) ).

% minNullmin
tff(fact_2111_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T) = none_nat )
     => pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T)) ) ).

% minminNull
tff(fact_2112_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,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw) )
               => ( ( Y = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy,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)),Ux,Uy,Uz2))) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_2113_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,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw) )
               => ( ( Y = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy,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)),Ux,Uy,Uz2))) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_2114_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_2115_inthall,axiom,
    ! [Xs: list_complex,P2: fun_complex_bool,N: nat] :
      ( ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),set_complex2(Xs)))
         => pp(aa_complex_bool(P2,X3)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3451745648224563538omplex(Xs)))
       => pp(aa_complex_bool(P2,aa_nat_complex(nth_complex(Xs),N))) ) ) ).

% inthall
tff(fact_2116_inthall,axiom,
    ! [Xs: list_real,P2: fun_real_bool,N: nat] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),set_real2(Xs)))
         => pp(aa_real_bool(P2,X3)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
       => pp(aa_real_bool(P2,aa_nat_real(nth_real(Xs),N))) ) ) ).

% inthall
tff(fact_2117_inthall,axiom,
    ! [Xs: list_set_nat,P2: fun_set_nat_bool,N: nat] :
      ( ! [X3: set_nat] :
          ( member_set_nat(X3,set_set_nat2(Xs))
         => pp(aa_set_nat_bool(P2,X3)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
       => pp(aa_set_nat_bool(P2,aa_nat_set_nat(nth_set_nat(Xs),N))) ) ) ).

% inthall
tff(fact_2118_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P2: fun_VEBT_VEBT_bool,N: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P2,X3)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
       => pp(aa_VEBT_VEBT_bool(P2,nth_VEBT_VEBT(Xs,N))) ) ) ).

% inthall
tff(fact_2119_inthall,axiom,
    ! [Xs: list_bool,P2: fun_bool_bool,N: nat] :
      ( ! [X3: bool] :
          ( member_bool(X3,set_bool2(Xs))
         => pp(aa_bool_bool(P2,X3)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
       => pp(aa_bool_bool(P2,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% inthall
tff(fact_2120_inthall,axiom,
    ! [Xs: list_nat,P2: fun_nat_bool,N: nat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
         => pp(aa_nat_bool(P2,X3)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
       => pp(aa_nat_bool(P2,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% inthall
tff(fact_2121_inthall,axiom,
    ! [Xs: list_int,P2: fun_int_bool,N: nat] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
         => pp(aa_int_bool(P2,X3)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
       => pp(aa_int_bool(P2,aa_nat_int(nth_int(Xs),N))) ) ) ).

% inthall
tff(fact_2122_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_12)) ) ).

% not_min_Null_member
tff(fact_2123_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_2124_Suc__diff__diff,axiom,
    ! [M2: nat,N: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),N)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),K2) ).

% Suc_diff_diff
tff(fact_2125_diff__Suc__Suc,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) ).

% diff_Suc_Suc
tff(fact_2126_diff__0__eq__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_2127_diff__self__eq__0,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),M2) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_2128_diff__diff__cancel,axiom,
    ! [I2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I2)) = I2 ) ) ).

% diff_diff_cancel
tff(fact_2129_diff__diff__left,axiom,
    ! [I2: nat,J2: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J2)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2)) ).

% diff_diff_left
tff(fact_2130_zero__less__diff,axiom,
    ! [N: nat,M2: 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,N),M2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% zero_less_diff
tff(fact_2131_diff__is__0__eq_H,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_2132_diff__is__0__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% diff_is_0_eq
tff(fact_2133_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J2)),K2) ) ) ).

% Nat.add_diff_assoc
tff(fact_2134_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2)),I2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),I2)),K2) ) ) ).

% Nat.add_diff_assoc2
tff(fact_2135_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),J2) ) ) ).

% Nat.diff_diff_right
tff(fact_2136_diff__Suc__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),one_one_nat) = N ).

% diff_Suc_1
tff(fact_2137_Suc__pred,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% Suc_pred
tff(fact_2138_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(suc,J2)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_2139_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J2))
     => ( 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,J2),K2))),I2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),I2)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_2140_Suc__diff__1,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ) ).

% Suc_diff_1
tff(fact_2141_zero__induct__lemma,axiom,
    ! [P2: fun_nat_bool,K2: nat,I2: nat] :
      ( pp(aa_nat_bool(P2,K2))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(P2,aa_nat_nat(suc,N3)))
           => pp(aa_nat_bool(P2,N3)) )
       => pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),I2))) ) ) ).

% zero_induct_lemma
tff(fact_2142_minus__nat_Odiff__0,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),zero_zero_nat) = M2 ).

% minus_nat.diff_0
tff(fact_2143_diffs0__imp__equal,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2) = zero_zero_nat )
       => ( M2 = N ) ) ) ).

% diffs0_imp_equal
tff(fact_2144_diff__less__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),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),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M2))) ) ) ).

% diff_less_mono2
tff(fact_2145_less__imp__diff__less,axiom,
    ! [J2: nat,K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),K2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),N)),K2)) ) ).

% less_imp_diff_less
tff(fact_2146_diff__le__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => 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),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M2))) ) ).

% diff_le_mono2
tff(fact_2147_le__diff__iff_H,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),B)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ) ).

% le_diff_iff'
tff(fact_2148_diff__le__self,axiom,
    ! [M2: nat,N: 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,M2),N)),M2)) ).

% diff_le_self
tff(fact_2149_diff__le__mono,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),L))) ) ).

% diff_le_mono
tff(fact_2150_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_2151_le__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ) ).

% le_diff_iff
tff(fact_2152_eq__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2) )
        <=> ( M2 = N ) ) ) ) ).

% eq_diff_iff
tff(fact_2153_Nat_Odiff__cancel,axiom,
    ! [K2: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) ).

% Nat.diff_cancel
tff(fact_2154_diff__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) ).

% diff_cancel2
tff(fact_2155_diff__add__inverse,axiom,
    ! [N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)),N) = M2 ).

% diff_add_inverse
tff(fact_2156_diff__add__inverse2,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)),N) = M2 ).

% diff_add_inverse2
tff(fact_2157_diff__mult__distrib,axiom,
    ! [M2: nat,N: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K2)) ).

% diff_mult_distrib
tff(fact_2158_diff__mult__distrib2,axiom,
    ! [K2: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)) ).

% diff_mult_distrib2
tff(fact_2159_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_2160_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_2161_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_2162_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [I: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
           => ( nth_VEBT_VEBT(Xs,I) = nth_VEBT_VEBT(Ys,I) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2163_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
        & ! [I: 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(Xs),I))
            <=> pp(aa_nat_bool(nth_bool(Ys),I)) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2164_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
        & ! [I: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I) = aa_nat_nat(nth_nat(Ys),I) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2165_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [I: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
           => ( aa_nat_int(nth_int(Xs),I) = aa_nat_int(nth_int(Ys),I) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2166_Skolem__list__nth,axiom,
    ! [K2: nat,P2: fun_na2351039247863542899T_bool] :
      ( ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K2))
         => ? [X_13: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P2,I),X_13)) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K2 )
          & ! [I: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K2))
             => pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P2,I),nth_VEBT_VEBT(Xs3,I))) ) ) ) ).

% Skolem_list_nth
tff(fact_2167_Skolem__list__nth,axiom,
    ! [K2: nat,P2: fun_na9137082442855928016l_bool] :
      ( ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K2))
         => ? [X_13: bool] : pp(aa_bool_bool(aa_nat_fun_bool_bool(P2,I),X_13)) )
    <=> ? [Xs3: list_bool] :
          ( ( size_size_list_bool(Xs3) = K2 )
          & ! [I: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K2))
             => pp(aa_bool_bool(aa_nat_fun_bool_bool(P2,I),aa_nat_bool(nth_bool(Xs3),I))) ) ) ) ).

% Skolem_list_nth
tff(fact_2168_Skolem__list__nth,axiom,
    ! [K2: nat,P2: fun_nat_fun_nat_bool] :
      ( ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K2))
         => ? [X_13: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,I),X_13)) )
    <=> ? [Xs3: list_nat] :
          ( ( aa_list_nat_nat(size_size_list_nat,Xs3) = K2 )
          & ! [I: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,I),aa_nat_nat(nth_nat(Xs3),I))) ) ) ) ).

% Skolem_list_nth
tff(fact_2169_Skolem__list__nth,axiom,
    ! [K2: nat,P2: fun_nat_fun_int_bool] :
      ( ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K2))
         => ? [X_13: int] : pp(aa_int_bool(aa_nat_fun_int_bool(P2,I),X_13)) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K2 )
          & ! [I: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K2))
             => pp(aa_int_bool(aa_nat_fun_int_bool(P2,I),aa_nat_int(nth_int(Xs3),I))) ) ) ) ).

% Skolem_list_nth
tff(fact_2170_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [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(Ys,I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_2171_nth__equalityI,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
     => ( ! [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(Ys),I3)) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_2172_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_2173_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [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) = aa_nat_int(nth_int(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_2174_diff__less__Suc,axiom,
    ! [M2: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),aa_nat_nat(suc,M2))) ).

% diff_less_Suc
tff(fact_2175_Suc__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) ) ) ).

% Suc_diff_Suc
tff(fact_2176_diff__less,axiom,
    ! [N: nat,M2: 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(ord_less_nat,zero_zero_nat),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),M2)) ) ) ).

% diff_less
tff(fact_2177_Suc__diff__le,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) ) ) ).

% Suc_diff_le
tff(fact_2178_less__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ) ).

% less_diff_iff
tff(fact_2179_diff__less__mono,axiom,
    ! [A: nat,B: nat,C2: 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,C2),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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C2))) ) ) ).

% diff_less_mono
tff(fact_2180_diff__add__0,axiom,
    ! [N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)) = zero_zero_nat ).

% diff_add_0
tff(fact_2181_finite__maxlen,axiom,
    ! [M5: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M5)
     => ? [N3: nat] :
        ! [X5: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X5,M5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(X5)),N3)) ) ) ).

% finite_maxlen
tff(fact_2182_finite__maxlen,axiom,
    ! [M5: set_list_bool] :
      ( finite7866827932528546246t_bool(M5)
     => ? [N3: nat] :
        ! [X5: list_bool] :
          ( member_list_bool(X5,M5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(X5)),N3)) ) ) ).

% finite_maxlen
tff(fact_2183_finite__maxlen,axiom,
    ! [M5: set_list_nat] :
      ( finite8100373058378681591st_nat(M5)
     => ? [N3: nat] :
        ! [X5: list_nat] :
          ( member_list_nat(X5,M5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_nat_nat(size_size_list_nat,X5)),N3)) ) ) ).

% finite_maxlen
tff(fact_2184_finite__maxlen,axiom,
    ! [M5: set_list_int] :
      ( finite3922522038869484883st_int(M5)
     => ? [N3: nat] :
        ! [X5: list_int] :
          ( member_list_int(X5,M5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(X5)),N3)) ) ) ).

% finite_maxlen
tff(fact_2185_less__diff__conv,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),J2)) ) ).

% less_diff_conv
tff(fact_2186_add__diff__inverse__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = M2 ) ) ).

% add_diff_inverse_nat
tff(fact_2187_le__diff__conv,axiom,
    ! [J2: nat,K2: nat,I2: 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,J2),K2)),I2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2))) ) ).

% le_diff_conv
tff(fact_2188_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),J2)) ) ) ).

% Nat.le_diff_conv2
tff(fact_2189_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J2)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2)) ) ) ).

% Nat.diff_add_assoc
tff(fact_2190_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),I2)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2)),I2) ) ) ).

% Nat.diff_add_assoc2
tff(fact_2191_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I2) = K2 )
      <=> ( J2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),I2) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_2192_diff__Suc__eq__diff__pred,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)),N) ).

% diff_Suc_eq_diff_pred
tff(fact_2193_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va3: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va3,Vb,Vc))) ).

% VEBT_internal.minNull.simps(5)
tff(fact_2194_nth__mem,axiom,
    ! [N: nat,Xs: list_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3451745648224563538omplex(Xs)))
     => pp(aa_set_complex_bool(member_complex(aa_nat_complex(nth_complex(Xs),N)),set_complex2(Xs))) ) ).

% nth_mem
tff(fact_2195_nth__mem,axiom,
    ! [N: nat,Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => pp(aa_set_real_bool(member_real(aa_nat_real(nth_real(Xs),N)),set_real2(Xs))) ) ).

% nth_mem
tff(fact_2196_nth__mem,axiom,
    ! [N: nat,Xs: list_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
     => member_set_nat(aa_nat_set_nat(nth_set_nat(Xs),N),set_set_nat2(Xs)) ) ).

% nth_mem
tff(fact_2197_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,nth_VEBT_VEBT(Xs,N)),set_VEBT_VEBT2(Xs))) ) ).

% nth_mem
tff(fact_2198_nth__mem,axiom,
    ! [N: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => member_bool(aa_nat_bool(nth_bool(Xs),N),set_bool2(Xs)) ) ).

% nth_mem
tff(fact_2199_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
     => pp(aa_set_nat_bool(member_nat(aa_nat_nat(nth_nat(Xs),N)),set_nat2(Xs))) ) ).

% nth_mem
tff(fact_2200_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => pp(aa_set_int_bool(member_int(aa_nat_int(nth_int(Xs),N)),set_int2(Xs))) ) ).

% nth_mem
tff(fact_2201_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,P2: fun_VEBT_VEBT_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( ! [X3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
           => pp(aa_VEBT_VEBT_bool(P2,X3)) )
       => pp(aa_VEBT_VEBT_bool(P2,nth_VEBT_VEBT(Xs,N))) ) ) ).

% list_ball_nth
tff(fact_2202_list__ball__nth,axiom,
    ! [N: nat,Xs: list_bool,P2: fun_bool_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => ( ! [X3: bool] :
            ( member_bool(X3,set_bool2(Xs))
           => pp(aa_bool_bool(P2,X3)) )
       => pp(aa_bool_bool(P2,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_2203_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P2: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
           => pp(aa_nat_bool(P2,X3)) )
       => pp(aa_nat_bool(P2,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_2204_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P2: fun_int_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => ( ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
           => pp(aa_int_bool(P2,X3)) )
       => pp(aa_int_bool(P2,aa_nat_int(nth_int(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_2205_in__set__conv__nth,axiom,
    ! [X: complex,Xs: list_complex] :
      ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s3451745648224563538omplex(Xs)))
          & ( aa_nat_complex(nth_complex(Xs),I) = X ) ) ) ).

% in_set_conv_nth
tff(fact_2206_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
          & ( aa_nat_real(nth_real(Xs),I) = X ) ) ) ).

% in_set_conv_nth
tff(fact_2207_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( member_set_nat(X,set_set_nat2(Xs))
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s3254054031482475050et_nat(Xs)))
          & ( aa_nat_set_nat(nth_set_nat(Xs),I) = X ) ) ) ).

% in_set_conv_nth
tff(fact_2208_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)))
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
          & ( nth_VEBT_VEBT(Xs,I) = X ) ) ) ).

% in_set_conv_nth
tff(fact_2209_in__set__conv__nth,axiom,
    ! [X: bool,Xs: list_bool] :
      ( member_bool(X,set_bool2(Xs))
    <=> ? [I: 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(Xs),I))
          <=> pp(X) ) ) ) ).

% in_set_conv_nth
tff(fact_2210_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
          & ( aa_nat_nat(nth_nat(Xs),I) = X ) ) ) ).

% in_set_conv_nth
tff(fact_2211_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
          & ( aa_nat_int(nth_int(Xs),I) = X ) ) ) ).

% in_set_conv_nth
tff(fact_2212_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P2: fun_complex_bool,X: complex] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s3451745648224563538omplex(Xs)))
         => pp(aa_complex_bool(P2,aa_nat_complex(nth_complex(Xs),I3))) )
     => ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
       => pp(aa_complex_bool(P2,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_2213_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P2: fun_real_bool,X: real] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_real(Xs)))
         => pp(aa_real_bool(P2,aa_nat_real(nth_real(Xs),I3))) )
     => ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
       => pp(aa_real_bool(P2,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_2214_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P2: fun_set_nat_bool,X: set_nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s3254054031482475050et_nat(Xs)))
         => pp(aa_set_nat_bool(P2,aa_nat_set_nat(nth_set_nat(Xs),I3))) )
     => ( member_set_nat(X,set_set_nat2(Xs))
       => pp(aa_set_nat_bool(P2,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_2215_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P2: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P2,nth_VEBT_VEBT(Xs,I3))) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
       => pp(aa_VEBT_VEBT_bool(P2,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_2216_all__nth__imp__all__set,axiom,
    ! [Xs: list_bool,P2: fun_bool_bool,X: bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P2,aa_nat_bool(nth_bool(Xs),I3))) )
     => ( member_bool(X,set_bool2(Xs))
       => pp(aa_bool_bool(P2,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_2217_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P2: fun_nat_bool,X: nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs)))
         => pp(aa_nat_bool(P2,aa_nat_nat(nth_nat(Xs),I3))) )
     => ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
       => pp(aa_nat_bool(P2,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_2218_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P2: fun_int_bool,X: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_int(Xs)))
         => pp(aa_int_bool(P2,aa_nat_int(nth_int(Xs),I3))) )
     => ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
       => pp(aa_int_bool(P2,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_2219_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P2: fun_VEBT_VEBT_bool] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P2,X4)) )
    <=> ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P2,nth_VEBT_VEBT(Xs,I))) ) ) ).

% all_set_conv_all_nth
tff(fact_2220_all__set__conv__all__nth,axiom,
    ! [Xs: list_bool,P2: fun_bool_bool] :
      ( ! [X4: bool] :
          ( member_bool(X4,set_bool2(Xs))
         => pp(aa_bool_bool(P2,X4)) )
    <=> ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P2,aa_nat_bool(nth_bool(Xs),I))) ) ) ).

% all_set_conv_all_nth
tff(fact_2221_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P2: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => pp(aa_nat_bool(P2,X4)) )
    <=> ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
         => pp(aa_nat_bool(P2,aa_nat_nat(nth_nat(Xs),I))) ) ) ).

% all_set_conv_all_nth
tff(fact_2222_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P2: fun_int_bool] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
         => pp(aa_int_bool(P2,X4)) )
    <=> ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
         => pp(aa_int_bool(P2,aa_nat_int(nth_int(Xs),I))) ) ) ).

% all_set_conv_all_nth
tff(fact_2223_diff__Suc__less,axiom,
    ! [N: nat,I2: 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(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,I2))),N)) ) ).

% diff_Suc_less
tff(fact_2224_nat__diff__split,axiom,
    ! [P2: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P2,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(P2,zero_zero_nat)) )
        & ! [D2: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2) )
           => pp(aa_nat_bool(P2,D2)) ) ) ) ).

% nat_diff_split
tff(fact_2225_nat__diff__split__asm,axiom,
    ! [P2: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P2,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(P2,zero_zero_nat)) )
          | ? [D2: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2) )
              & ~ pp(aa_nat_bool(P2,D2)) ) ) ) ).

% nat_diff_split_asm
tff(fact_2226_less__diff__conv2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),J2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2)),I2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2))) ) ) ).

% less_diff_conv2
tff(fact_2227_nat__eq__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),I2))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N) )
      <=> ( 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,I2),J2)),U)),M2) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_2228_nat__eq__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N) )
      <=> ( M2 = 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,J2),I2)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_2229_nat__le__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),I2))
     => ( 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,I2),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),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(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J2)),U)),M2)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_2230_nat__le__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M2: nat,N: 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_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),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,J2),I2)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_2231_nat__diff__add__eq1,axiom,
    ! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),I2))
     => ( 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,I2),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N)) = 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,I2),J2)),U)),M2)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_2232_nat__diff__add__eq2,axiom,
    ! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
     => ( 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,I2),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),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,J2),I2)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_2233_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,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_2234_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( ( X != vEBT_Leaf(fFalse,fFalse) )
       => ~ ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_2235_Suc__pred_H,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_2236_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_2237_add__eq__if,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N) = N ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N) = 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,M2),one_one_nat)),N)) ) ) ) ).

% add_eq_if
tff(fact_2238_nat__less__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),I2))
     => ( 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,I2),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N)))
      <=> 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,I2),J2)),U)),M2)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_2239_nat__less__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M2: nat,N: 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,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),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,J2),I2)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_2240_mult__eq__if,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) = zero_zero_nat ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)),N)) ) ) ) ).

% mult_eq_if
tff(fact_2241_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) )
           => ( ( ? [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy)
               => ~ pp(Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2)
                 => pp(Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_2242_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: 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),D22))
       => ? [E2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E2),D1))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E2),D22)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_2243_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: 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),D22))
       => ? [E2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E2),D1))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E2),D22)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_2244_finite__has__minimal2,axiom,
    ! [A4: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(A),A4))
       => ? [X3: real] :
            ( pp(aa_set_real_bool(member_real(X3),A4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),A))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X3))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_2245_finite__has__minimal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( member_set_nat(A,A4)
       => ? [X3: set_nat] :
            ( member_set_nat(X3,A4)
            & pp(aa_set_nat_bool(ord_less_eq_set_nat(X3),A))
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A4)
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Xa),X3))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_2246_finite__has__minimal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( finite_finite_rat(A4)
     => ( member_rat(A,A4)
       => ? [X3: rat] :
            ( member_rat(X3,A4)
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),A))
            & ! [Xa: rat] :
                ( member_rat(Xa,A4)
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa),X3))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_2247_finite__has__minimal2,axiom,
    ! [A4: set_num,A: num] :
      ( finite_finite_num(A4)
     => ( member_num(A,A4)
       => ? [X3: num] :
            ( member_num(X3,A4)
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),A))
            & ! [Xa: num] :
                ( member_num(Xa,A4)
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa),X3))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_2248_finite__has__minimal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(member_nat(A),A4))
       => ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A4))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),A))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A4))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X3))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_2249_finite__has__minimal2,axiom,
    ! [A4: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(A),A4))
       => ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A4))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),A))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A4))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X3))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_2250_finite__has__maximal2,axiom,
    ! [A4: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(A),A4))
       => ? [X3: real] :
            ( pp(aa_set_real_bool(member_real(X3),A4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
            & ! [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),A4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Xa))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_2251_finite__has__maximal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( member_set_nat(A,A4)
       => ? [X3: set_nat] :
            ( member_set_nat(X3,A4)
            & pp(aa_set_nat_bool(ord_less_eq_set_nat(A),X3))
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A4)
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X3),Xa))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_2252_finite__has__maximal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( finite_finite_rat(A4)
     => ( member_rat(A,A4)
       => ? [X3: rat] :
            ( member_rat(X3,A4)
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),X3))
            & ! [Xa: rat] :
                ( member_rat(Xa,A4)
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Xa))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_2253_finite__has__maximal2,axiom,
    ! [A4: set_num,A: num] :
      ( finite_finite_num(A4)
     => ( member_num(A,A4)
       => ? [X3: num] :
            ( member_num(X3,A4)
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),X3))
            & ! [Xa: num] :
                ( member_num(Xa,A4)
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Xa))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_2254_finite__has__maximal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(member_nat(A),A4))
       => ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A4))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X3))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A4))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Xa))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_2255_finite__has__maximal2,axiom,
    ! [A4: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(A),A4))
       => ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A4))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X3))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A4))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Xa))
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_2256_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))) ) )
             => ( ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy) )
                   => ( pp(Y)
                     => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy))) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) )
                     => ( ~ pp(Y)
                       => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_2257_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))) )
         => ~ ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy))) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_2258_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,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2))) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_2259_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_VEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_Pr744662078594809490T_VEBT(enumerate_VEBT_VEBT(N,Xs),M2) = produc599794634098209291T_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2),nth_VEBT_VEBT(Xs,M2)) ) ) ).

% nth_enumerate_eq
tff(fact_2260_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_bool,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),size_size_list_bool(Xs)))
     => ( nth_Pr396318242284072693t_bool(enumerate_bool(N,Xs),M2) = produc3913204875133826030t_bool(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2),aa_nat_bool(nth_bool(Xs),M2)) ) ) ).

% nth_enumerate_eq
tff(fact_2261_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( nth_Pr7617993195940197384at_nat(enumerate_nat(N,Xs),M2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)),aa_nat_nat(nth_nat(Xs),M2)) ) ) ).

% nth_enumerate_eq
tff(fact_2262_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),size_size_list_int(Xs)))
     => ( nth_Pr3440142176431000676at_int(enumerate_int(N,Xs),M2) = product_Pair_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2),aa_nat_int(nth_int(Xs),M2)) ) ) ).

% nth_enumerate_eq
tff(fact_2263_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F2: 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(F2,lattic4729654577720512673ex_rat(F2,S3))),aa_complex_rat(F2,Y))) ) ) ) ).

% arg_min_least
tff(fact_2264_arg__min__least,axiom,
    ! [S3: set_nat,Y: nat,F2: 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(F2,aa_set_nat_nat(lattic6811802900495863747at_rat(F2),S3))),aa_nat_rat(F2,Y))) ) ) ) ).

% arg_min_least
tff(fact_2265_arg__min__least,axiom,
    ! [S3: set_int,Y: int,F2: 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(F2,lattic7811156612396918303nt_rat(F2,S3))),aa_int_rat(F2,Y))) ) ) ) ).

% arg_min_least
tff(fact_2266_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( S3 != bot_bot_set_real )
       => ( pp(aa_set_real_bool(member_real(Y),S3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F2,lattic4420706379359479199al_rat(F2,S3))),aa_real_rat(F2,Y))) ) ) ) ).

% arg_min_least
tff(fact_2267_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F2: 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(F2,lattic1922116423962787043ex_num(F2,S3))),aa_complex_num(F2,Y))) ) ) ) ).

% arg_min_least
tff(fact_2268_arg__min__least,axiom,
    ! [S3: set_nat,Y: nat,F2: 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(F2,aa_set_nat_nat(lattic4004264746738138117at_num(F2),S3))),aa_nat_num(F2,Y))) ) ) ) ).

% arg_min_least
tff(fact_2269_arg__min__least,axiom,
    ! [S3: set_int,Y: int,F2: 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(F2,lattic5003618458639192673nt_num(F2,S3))),aa_int_num(F2,Y))) ) ) ) ).

% arg_min_least
tff(fact_2270_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F2: fun_real_num] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( S3 != bot_bot_set_real )
       => ( pp(aa_set_real_bool(member_real(Y),S3))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_real_num(F2,lattic1613168225601753569al_num(F2,S3))),aa_real_num(F2,Y))) ) ) ) ).

% arg_min_least
tff(fact_2271_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( S3 != bot_bot_set_complex )
       => ( pp(aa_set_complex_bool(member_complex(Y),S3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F2,lattic5364784637807008409ex_nat(F2,S3))),aa_complex_nat(F2,Y))) ) ) ) ).

% arg_min_least
tff(fact_2272_arg__min__least,axiom,
    ! [S3: set_nat,Y: nat,F2: fun_nat_nat] :
      ( 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_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F2,aa_set_nat_nat(lattic7446932960582359483at_nat(F2),S3))),aa_nat_nat(F2,Y))) ) ) ) ).

% arg_min_least
tff(fact_2273_Euclid__induct,axiom,
    ! [P2: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A3),B3))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,B3),A3)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A3),zero_zero_nat))
       => ( ! [A3: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A3),B3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,A),B)) ) ) ) ).

% Euclid_induct
tff(fact_2274_nat__descend__induct,axiom,
    ! [N: nat,P2: fun_nat_bool,M2: nat] :
      ( ! [K: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
         => pp(aa_nat_bool(P2,K)) )
     => ( ! [K: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
           => ( ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),I4))
                 => pp(aa_nat_bool(P2,I4)) )
             => pp(aa_nat_bool(P2,K)) ) )
       => pp(aa_nat_bool(P2,M2)) ) ) ).

% nat_descend_induct
tff(fact_2275_length__mul__elem,axiom,
    ! [Xs: list_list_VEBT_VEBT,N: nat] :
      ( ! [X3: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X3,set_list_VEBT_VEBT2(Xs))
         => ( size_s6755466524823107622T_VEBT(X3) = N ) )
     => ( size_s6755466524823107622T_VEBT(concat_VEBT_VEBT(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s8217280938318005548T_VEBT(Xs)),N) ) ) ).

% length_mul_elem
tff(fact_2276_length__mul__elem,axiom,
    ! [Xs: list_list_bool,N: nat] :
      ( ! [X3: list_bool] :
          ( member_list_bool(X3,set_list_bool2(Xs))
         => ( size_size_list_bool(X3) = N ) )
     => ( size_size_list_bool(concat_bool(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s9193230429364170767t_bool(Xs)),N) ) ) ).

% length_mul_elem
tff(fact_2277_length__mul__elem,axiom,
    ! [Xs: list_list_nat,N: nat] :
      ( ! [X3: list_nat] :
          ( member_list_nat(X3,set_list_nat2(Xs))
         => ( aa_list_nat_nat(size_size_list_nat,X3) = N ) )
     => ( aa_list_nat_nat(size_size_list_nat,concat_nat(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3023201423986296836st_nat(Xs)),N) ) ) ).

% length_mul_elem
tff(fact_2278_length__mul__elem,axiom,
    ! [Xs: list_list_int,N: nat] :
      ( ! [X3: list_int] :
          ( member_list_int(X3,set_list_int2(Xs))
         => ( size_size_list_int(X3) = N ) )
     => ( size_size_list_int(concat_int(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s533118279054570080st_int(Xs)),N) ) ) ).

% length_mul_elem
tff(fact_2279_triangle__Suc,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,N)),aa_nat_nat(suc,N)) ).

% triangle_Suc
tff(fact_2280_length__enumerate,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(enumerate_VEBT_VEBT(N,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_enumerate
tff(fact_2281_length__enumerate,axiom,
    ! [N: nat,Xs: list_bool] : size_s783543119881238752t_bool(enumerate_bool(N,Xs)) = size_size_list_bool(Xs) ).

% length_enumerate
tff(fact_2282_length__enumerate,axiom,
    ! [N: nat,Xs: list_nat] : size_s5460976970255530739at_nat(enumerate_nat(N,Xs)) = aa_list_nat_nat(size_size_list_nat,Xs) ).

% length_enumerate
tff(fact_2283_length__enumerate,axiom,
    ! [N: nat,Xs: list_int] : size_s2970893825323803983at_int(enumerate_int(N,Xs)) = size_size_list_int(Xs) ).

% length_enumerate
tff(fact_2284_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_2285_diff__commute,axiom,
    ! [I2: nat,J2: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J2)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),K2)),J2) ).

% diff_commute
tff(fact_2286_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_2287_prod__decode__aux_Osimps,axiom,
    ! [M2: nat,K2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),K2))
       => ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K2),M2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),M2)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),K2))
       => ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K2),M2) = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,K2))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_2288_prod__encode__prod__decode__aux,axiom,
    ! [K2: nat,M2: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K2),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K2)),M2) ).

% prod_encode_prod_decode_aux
tff(fact_2289_prod__decode__triangle__add,axiom,
    ! [K2: nat,M2: nat] : aa_nat7603221163087113075at_nat(nat_prod_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K2)),M2)) = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K2),M2) ).

% prod_decode_triangle_add
tff(fact_2290_in__set__product__lists__length,axiom,
    ! [Xs: list_VEBT_VEBT,Xss: list_list_VEBT_VEBT] :
      ( member2936631157270082147T_VEBT(Xs,set_list_VEBT_VEBT2(produc3021084454716106787T_VEBT(Xss)))
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s8217280938318005548T_VEBT(Xss) ) ) ).

% in_set_product_lists_length
tff(fact_2291_in__set__product__lists__length,axiom,
    ! [Xs: list_bool,Xss: list_list_bool] :
      ( member_list_bool(Xs,set_list_bool2(product_lists_bool(Xss)))
     => ( size_size_list_bool(Xs) = size_s9193230429364170767t_bool(Xss) ) ) ).

% in_set_product_lists_length
tff(fact_2292_in__set__product__lists__length,axiom,
    ! [Xs: list_nat,Xss: list_list_nat] :
      ( member_list_nat(Xs,set_list_nat2(product_lists_nat(Xss)))
     => ( aa_list_nat_nat(size_size_list_nat,Xs) = size_s3023201423986296836st_nat(Xss) ) ) ).

% in_set_product_lists_length
tff(fact_2293_in__set__product__lists__length,axiom,
    ! [Xs: list_int,Xss: list_list_int] :
      ( member_list_int(Xs,set_list_int2(product_lists_int(Xss)))
     => ( size_size_list_int(Xs) = size_s533118279054570080st_int(Xss) ) ) ).

% in_set_product_lists_length
tff(fact_2294_Gcd__remove0__nat,axiom,
    ! [M5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M5))
     => ( aa_set_nat_nat(gcd_Gcd_nat,M5) = aa_set_nat_nat(gcd_Gcd_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,M5),insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_2295_inf__period_I2_J,axiom,
    ! [P2: fun_real_bool,D3: real,Q: fun_real_bool] :
      ( ! [X3: real,K: real] :
          ( pp(aa_real_bool(P2,X3))
        <=> pp(aa_real_bool(P2,aa_real_real(minus_minus_real(X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K),D3)))) )
     => ( ! [X3: real,K: real] :
            ( pp(aa_real_bool(Q,X3))
          <=> pp(aa_real_bool(Q,aa_real_real(minus_minus_real(X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K),D3)))) )
       => ! [X5: real,K4: real] :
            ( ( pp(aa_real_bool(P2,X5))
              | pp(aa_real_bool(Q,X5)) )
          <=> ( pp(aa_real_bool(P2,aa_real_real(minus_minus_real(X5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3))))
              | pp(aa_real_bool(Q,aa_real_real(minus_minus_real(X5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3)))) ) ) ) ) ).

% inf_period(2)
tff(fact_2296_inf__period_I2_J,axiom,
    ! [P2: fun_rat_bool,D3: rat,Q: fun_rat_bool] :
      ( ! [X3: rat,K: rat] :
          ( pp(aa_rat_bool(P2,X3))
        <=> pp(aa_rat_bool(P2,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D3)))) )
     => ( ! [X3: rat,K: rat] :
            ( pp(aa_rat_bool(Q,X3))
          <=> pp(aa_rat_bool(Q,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D3)))) )
       => ! [X5: rat,K4: rat] :
            ( ( pp(aa_rat_bool(P2,X5))
              | pp(aa_rat_bool(Q,X5)) )
          <=> ( pp(aa_rat_bool(P2,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3))))
              | pp(aa_rat_bool(Q,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3)))) ) ) ) ) ).

% inf_period(2)
tff(fact_2297_inf__period_I2_J,axiom,
    ! [P2: fun_int_bool,D3: int,Q: fun_int_bool] :
      ( ! [X3: int,K: int] :
          ( pp(aa_int_bool(P2,X3))
        <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3)))) )
     => ( ! [X3: int,K: int] :
            ( pp(aa_int_bool(Q,X3))
          <=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3)))) )
       => ! [X5: int,K4: int] :
            ( ( pp(aa_int_bool(P2,X5))
              | pp(aa_int_bool(Q,X5)) )
          <=> ( pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))))
              | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3)))) ) ) ) ) ).

% inf_period(2)
tff(fact_2298_inf__period_I1_J,axiom,
    ! [P2: fun_real_bool,D3: real,Q: fun_real_bool] :
      ( ! [X3: real,K: real] :
          ( pp(aa_real_bool(P2,X3))
        <=> pp(aa_real_bool(P2,aa_real_real(minus_minus_real(X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K),D3)))) )
     => ( ! [X3: real,K: real] :
            ( pp(aa_real_bool(Q,X3))
          <=> pp(aa_real_bool(Q,aa_real_real(minus_minus_real(X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K),D3)))) )
       => ! [X5: real,K4: real] :
            ( ( pp(aa_real_bool(P2,X5))
              & pp(aa_real_bool(Q,X5)) )
          <=> ( pp(aa_real_bool(P2,aa_real_real(minus_minus_real(X5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3))))
              & pp(aa_real_bool(Q,aa_real_real(minus_minus_real(X5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3)))) ) ) ) ) ).

% inf_period(1)
tff(fact_2299_inf__period_I1_J,axiom,
    ! [P2: fun_rat_bool,D3: rat,Q: fun_rat_bool] :
      ( ! [X3: rat,K: rat] :
          ( pp(aa_rat_bool(P2,X3))
        <=> pp(aa_rat_bool(P2,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D3)))) )
     => ( ! [X3: rat,K: rat] :
            ( pp(aa_rat_bool(Q,X3))
          <=> pp(aa_rat_bool(Q,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D3)))) )
       => ! [X5: rat,K4: rat] :
            ( ( pp(aa_rat_bool(P2,X5))
              & pp(aa_rat_bool(Q,X5)) )
          <=> ( pp(aa_rat_bool(P2,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3))))
              & pp(aa_rat_bool(Q,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3)))) ) ) ) ) ).

% inf_period(1)
tff(fact_2300_inf__period_I1_J,axiom,
    ! [P2: fun_int_bool,D3: int,Q: fun_int_bool] :
      ( ! [X3: int,K: int] :
          ( pp(aa_int_bool(P2,X3))
        <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3)))) )
     => ( ! [X3: int,K: int] :
            ( pp(aa_int_bool(Q,X3))
          <=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3)))) )
       => ! [X5: int,K4: int] :
            ( ( pp(aa_int_bool(P2,X5))
              & pp(aa_int_bool(Q,X5)) )
          <=> ( pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))))
              & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3)))) ) ) ) ) ).

% inf_period(1)
tff(fact_2301_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_2302_verit__sum__simplify,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% verit_sum_simplify
tff(fact_2303_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_2304_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_2305_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_2306_Gcd__empty,axiom,
    aa_set_nat_nat(gcd_Gcd_nat,bot_bot_set_nat) = zero_zero_nat ).

% Gcd_empty
tff(fact_2307_Gcd__empty,axiom,
    gcd_Gcd_int(bot_bot_set_int) = zero_zero_int ).

% Gcd_empty
tff(fact_2308_Gcd__0__iff,axiom,
    ! [A4: set_int] :
      ( ( gcd_Gcd_int(A4) = zero_zero_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(A4),insert_int(zero_zero_int,bot_bot_set_int))) ) ).

% Gcd_0_iff
tff(fact_2309_Gcd__0__iff,axiom,
    ! [A4: set_nat] :
      ( ( aa_set_nat_nat(gcd_Gcd_nat,A4) = zero_zero_nat )
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(A4),insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ).

% Gcd_0_iff
tff(fact_2310_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_2311_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_2312_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_2313_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_2314_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_2315_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_2316_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_2317_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_2318_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_2319_prod__decode__def,axiom,
    nat_prod_decode = nat_prod_decode_aux(zero_zero_nat) ).

% prod_decode_def
tff(fact_2320_verit__comp__simplify1_I3_J,axiom,
    ! [B2: real,A2: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A2),B2)) ) ).

% verit_comp_simplify1(3)
tff(fact_2321_verit__comp__simplify1_I3_J,axiom,
    ! [B2: rat,A2: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A2),B2)) ) ).

% verit_comp_simplify1(3)
tff(fact_2322_verit__comp__simplify1_I3_J,axiom,
    ! [B2: num,A2: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2)) ) ).

% verit_comp_simplify1(3)
tff(fact_2323_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% verit_comp_simplify1(3)
tff(fact_2324_verit__comp__simplify1_I3_J,axiom,
    ! [B2: int,A2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% verit_comp_simplify1(3)
tff(fact_2325_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X5))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),T)) ) ).

% pinf(6)
tff(fact_2326_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X5))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),T)) ) ).

% pinf(6)
tff(fact_2327_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X5))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),T)) ) ).

% pinf(6)
tff(fact_2328_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X5))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),T)) ) ).

% pinf(6)
tff(fact_2329_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X5))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),T)) ) ).

% pinf(6)
tff(fact_2330_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X5))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X5)) ) ).

% pinf(8)
tff(fact_2331_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X5))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X5)) ) ).

% pinf(8)
tff(fact_2332_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X5))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X5)) ) ).

% pinf(8)
tff(fact_2333_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X5))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X5)) ) ).

% pinf(8)
tff(fact_2334_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X5))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X5)) ) ).

% pinf(8)
tff(fact_2335_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),T)) ) ).

% minf(6)
tff(fact_2336_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),T)) ) ).

% minf(6)
tff(fact_2337_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),T)) ) ).

% minf(6)
tff(fact_2338_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),T)) ) ).

% minf(6)
tff(fact_2339_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),T)) ) ).

% minf(6)
tff(fact_2340_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X5)) ) ).

% minf(8)
tff(fact_2341_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X5)) ) ).

% minf(8)
tff(fact_2342_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X5)) ) ).

% minf(8)
tff(fact_2343_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X5)) ) ).

% minf(8)
tff(fact_2344_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X5)) ) ).

% minf(8)
tff(fact_2345_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] : size_s7466405169056248089T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_2346_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_bool] : size_s1343096544515248124T_bool(produc5466675335756156456T_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_bool(Ys)) ).

% length_product
tff(fact_2347_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat] : size_s6152045936467909847BT_nat(produc7295137177222721919BT_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),aa_list_nat_nat(size_size_list_nat,Ys)) ).

% length_product
tff(fact_2348_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int] : size_s3661962791536183091BT_int(produc7292646706713671643BT_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_2349_length__product,axiom,
    ! [Xs: list_bool,Ys: list_VEBT_VEBT] : size_s7211631430880329078T_VEBT(produc2282853104913622818T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_2350_length__product,axiom,
    ! [Xs: list_bool,Ys: list_bool] : size_s6988337448923680857l_bool(product_bool_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_bool(Ys)) ).

% length_product
tff(fact_2351_length__product,axiom,
    ! [Xs: list_bool,Ys: list_nat] : size_s3243772863075988794ol_nat(product_bool_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),aa_list_nat_nat(size_size_list_nat,Ys)) ).

% length_product
tff(fact_2352_length__product,axiom,
    ! [Xs: list_bool,Ys: list_int] : size_s753689718144262038ol_int(product_bool_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_2353_length__product,axiom,
    ! [Xs: list_nat,Ys: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_2354_length__product,axiom,
    ! [Xs: list_nat,Ys: list_bool] : size_s783543119881238752t_bool(product_nat_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),size_size_list_bool(Ys)) ).

% length_product
tff(fact_2355_complete__interval,axiom,
    ! [A: real,B: real,P2: fun_real_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(P2,A))
       => ( ~ pp(aa_real_bool(P2,B))
         => ? [C: 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))
              & ! [X5: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X5))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),C)) )
                 => pp(aa_real_bool(P2,X5)) )
              & ! [D4: real] :
                  ( ! [X3: real] :
                      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
                        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),D4)) )
                     => pp(aa_real_bool(P2,X3)) )
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D4),C)) ) ) ) ) ) ).

% complete_interval
tff(fact_2356_complete__interval,axiom,
    ! [A: nat,B: nat,P2: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(P2,A))
       => ( ~ pp(aa_nat_bool(P2,B))
         => ? [C: 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,C),B))
              & ! [X5: nat] :
                  ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X5))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),C)) )
                 => pp(aa_nat_bool(P2,X5)) )
              & ! [D4: nat] :
                  ( ! [X3: nat] :
                      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X3))
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),D4)) )
                     => pp(aa_nat_bool(P2,X3)) )
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D4),C)) ) ) ) ) ) ).

% complete_interval
tff(fact_2357_complete__interval,axiom,
    ! [A: int,B: int,P2: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(P2,A))
       => ( ~ pp(aa_int_bool(P2,B))
         => ? [C: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
              & ! [X5: int] :
                  ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X5))
                    & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),C)) )
                 => pp(aa_int_bool(P2,X5)) )
              & ! [D4: int] :
                  ( ! [X3: int] :
                      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X3))
                        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),D4)) )
                     => pp(aa_int_bool(P2,X3)) )
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D4),C)) ) ) ) ) ) ).

% complete_interval
tff(fact_2358_vebt__insert_Osimps_I4_J,axiom,
    ! [V2: 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,V2)),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,V2)),TreeList2,Summary) ).

% vebt_insert.simps(4)
tff(fact_2359_nth__zip,axiom,
    ! [I2: nat,Xs: list_Code_integer,Ys: list_Code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s3445333598471063425nteger(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s3445333598471063425nteger(Ys)))
       => ( nth_Pr2304437835452373666nteger(zip_Co3543743374963494515nteger(Xs,Ys),I2) = produc1086072967326762835nteger(aa_nat_Code_integer(nth_Code_integer(Xs),I2),aa_nat_Code_integer(nth_Code_integer(Ys),I2)) ) ) ) ).

% nth_zip
tff(fact_2360_nth__zip,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Ys)))
       => ( nth_Pr4953567300277697838T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys),I2) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,I2),nth_VEBT_VEBT(Ys,I2)) ) ) ) ).

% nth_zip
tff(fact_2361_nth__zip,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,Ys: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Ys)))
       => ( nth_Pr1289831921885633937T_bool(zip_VEBT_VEBT_bool(Xs,Ys),I2) = produc3035007008238225530T_bool(nth_VEBT_VEBT(Xs,I2),aa_nat_bool(nth_bool(Ys),I2)) ) ) ) ).

% nth_zip
tff(fact_2362_nth__zip,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Ys)))
       => ( nth_Pr1791586995822124652BT_nat(zip_VEBT_VEBT_nat(Xs,Ys),I2) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,I2),aa_nat_nat(nth_nat(Ys),I2)) ) ) ) ).

% nth_zip
tff(fact_2363_nth__zip,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Ys)))
       => ( nth_Pr6837108013167703752BT_int(zip_VEBT_VEBT_int(Xs,Ys),I2) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,I2),aa_nat_int(nth_int(Ys),I2)) ) ) ) ).

% nth_zip
tff(fact_2364_nth__zip,axiom,
    ! [I2: nat,Xs: list_bool,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Ys)))
       => ( nth_Pr3250499561091541003T_VEBT(zip_bool_VEBT_VEBT(Xs,Ys),I2) = produc9074556814250467700T_VEBT(aa_nat_bool(nth_bool(Xs),I2),nth_VEBT_VEBT(Ys,I2)) ) ) ) ).

% nth_zip
tff(fact_2365_nth__zip,axiom,
    ! [I2: nat,Xs: list_bool,Ys: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Ys)))
       => ( nth_Pr2856333732135533678l_bool(zip_bool_bool(Xs,Ys),I2) = produc8297744024737662807l_bool(aa_nat_bool(nth_bool(Xs),I2),aa_nat_bool(nth_bool(Ys),I2)) ) ) ) ).

% nth_zip
tff(fact_2366_nth__zip,axiom,
    ! [I2: nat,Xs: list_bool,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Ys)))
       => ( nth_Pr2186422209488468047ol_nat(zip_bool_nat(Xs,Ys),I2) = produc8874003862241262736ol_nat(aa_nat_bool(nth_bool(Xs),I2),aa_nat_nat(nth_nat(Ys),I2)) ) ) ) ).

% nth_zip
tff(fact_2367_nth__zip,axiom,
    ! [I2: nat,Xs: list_bool,Ys: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Ys)))
       => ( nth_Pr7231943226834047147ol_int(zip_bool_int(Xs,Ys),I2) = produc8871513391732212460ol_int(aa_nat_bool(nth_bool(Xs),I2),aa_nat_int(nth_int(Ys),I2)) ) ) ) ).

% nth_zip
tff(fact_2368_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Ys)))
       => ( nth_Pr744662078594809490T_VEBT(zip_nat_VEBT_VEBT(Xs,Ys),I2) = produc599794634098209291T_VEBT(aa_nat_nat(nth_nat(Xs),I2),nth_VEBT_VEBT(Ys,I2)) ) ) ) ).

% nth_zip
tff(fact_2369_find__Some__iff2,axiom,
    ! [X: product_prod_nat_nat,P2: fun_Pr7757163693017731481t_bool,Xs: list_P6011104703257516679at_nat] :
      ( ( some_P7363390416028606310at_nat(X) = find_P8199882355184865565at_nat(P2,Xs) )
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s5460976970255530739at_nat(Xs)))
          & pp(aa_Pro2232559814647311250t_bool(P2,nth_Pr7617993195940197384at_nat(Xs,I)))
          & ( X = nth_Pr7617993195940197384at_nat(Xs,I) )
          & ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),I))
             => ~ pp(aa_Pro2232559814647311250t_bool(P2,nth_Pr7617993195940197384at_nat(Xs,J))) ) ) ) ).

% find_Some_iff2
tff(fact_2370_find__Some__iff2,axiom,
    ! [X: num,P2: fun_num_bool,Xs: list_num] :
      ( ( some_num(X) = find_num(P2,Xs) )
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_num(Xs)))
          & pp(aa_num_bool(P2,aa_nat_num(nth_num(Xs),I)))
          & ( X = aa_nat_num(nth_num(Xs),I) )
          & ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),I))
             => ~ pp(aa_num_bool(P2,aa_nat_num(nth_num(Xs),J))) ) ) ) ).

% find_Some_iff2
tff(fact_2371_find__Some__iff2,axiom,
    ! [X: vEBT_VEBT,P2: fun_VEBT_VEBT_bool,Xs: list_VEBT_VEBT] :
      ( ( some_VEBT_VEBT(X) = find_VEBT_VEBT(P2,Xs) )
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
          & pp(aa_VEBT_VEBT_bool(P2,nth_VEBT_VEBT(Xs,I)))
          & ( X = nth_VEBT_VEBT(Xs,I) )
          & ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),I))
             => ~ pp(aa_VEBT_VEBT_bool(P2,nth_VEBT_VEBT(Xs,J))) ) ) ) ).

% find_Some_iff2
tff(fact_2372_find__Some__iff2,axiom,
    ! [X: bool,P2: fun_bool_bool,Xs: list_bool] :
      ( ( some_bool(X) = find_bool(P2,Xs) )
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
          & pp(aa_bool_bool(P2,aa_nat_bool(nth_bool(Xs),I)))
          & ( pp(X)
          <=> pp(aa_nat_bool(nth_bool(Xs),I)) )
          & ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),I))
             => ~ pp(aa_bool_bool(P2,aa_nat_bool(nth_bool(Xs),J))) ) ) ) ).

% find_Some_iff2
tff(fact_2373_find__Some__iff2,axiom,
    ! [X: nat,P2: fun_nat_bool,Xs: list_nat] :
      ( ( some_nat(X) = find_nat(P2,Xs) )
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
          & pp(aa_nat_bool(P2,aa_nat_nat(nth_nat(Xs),I)))
          & ( X = aa_nat_nat(nth_nat(Xs),I) )
          & ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),I))
             => ~ pp(aa_nat_bool(P2,aa_nat_nat(nth_nat(Xs),J))) ) ) ) ).

% find_Some_iff2
tff(fact_2374_find__Some__iff2,axiom,
    ! [X: int,P2: fun_int_bool,Xs: list_int] :
      ( ( some_int(X) = find_int(P2,Xs) )
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
          & pp(aa_int_bool(P2,aa_nat_int(nth_int(Xs),I)))
          & ( X = aa_nat_int(nth_int(Xs),I) )
          & ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),I))
             => ~ pp(aa_int_bool(P2,aa_nat_int(nth_int(Xs),J))) ) ) ) ).

% find_Some_iff2
tff(fact_2375_find__Some__iff,axiom,
    ! [P2: fun_Pr7757163693017731481t_bool,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( ( find_P8199882355184865565at_nat(P2,Xs) = some_P7363390416028606310at_nat(X) )
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s5460976970255530739at_nat(Xs)))
          & pp(aa_Pro2232559814647311250t_bool(P2,nth_Pr7617993195940197384at_nat(Xs,I)))
          & ( X = nth_Pr7617993195940197384at_nat(Xs,I) )
          & ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),I))
             => ~ pp(aa_Pro2232559814647311250t_bool(P2,nth_Pr7617993195940197384at_nat(Xs,J))) ) ) ) ).

% find_Some_iff
tff(fact_2376_find__Some__iff,axiom,
    ! [P2: fun_num_bool,Xs: list_num,X: num] :
      ( ( find_num(P2,Xs) = some_num(X) )
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_num(Xs)))
          & pp(aa_num_bool(P2,aa_nat_num(nth_num(Xs),I)))
          & ( X = aa_nat_num(nth_num(Xs),I) )
          & ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),I))
             => ~ pp(aa_num_bool(P2,aa_nat_num(nth_num(Xs),J))) ) ) ) ).

% find_Some_iff
tff(fact_2377_find__Some__iff,axiom,
    ! [P2: fun_VEBT_VEBT_bool,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( find_VEBT_VEBT(P2,Xs) = some_VEBT_VEBT(X) )
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
          & pp(aa_VEBT_VEBT_bool(P2,nth_VEBT_VEBT(Xs,I)))
          & ( X = nth_VEBT_VEBT(Xs,I) )
          & ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),I))
             => ~ pp(aa_VEBT_VEBT_bool(P2,nth_VEBT_VEBT(Xs,J))) ) ) ) ).

% find_Some_iff
tff(fact_2378_find__Some__iff,axiom,
    ! [P2: fun_bool_bool,Xs: list_bool,X: bool] :
      ( ( find_bool(P2,Xs) = some_bool(X) )
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
          & pp(aa_bool_bool(P2,aa_nat_bool(nth_bool(Xs),I)))
          & ( pp(X)
          <=> pp(aa_nat_bool(nth_bool(Xs),I)) )
          & ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),I))
             => ~ pp(aa_bool_bool(P2,aa_nat_bool(nth_bool(Xs),J))) ) ) ) ).

% find_Some_iff
tff(fact_2379_find__Some__iff,axiom,
    ! [P2: fun_nat_bool,Xs: list_nat,X: nat] :
      ( ( find_nat(P2,Xs) = some_nat(X) )
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
          & pp(aa_nat_bool(P2,aa_nat_nat(nth_nat(Xs),I)))
          & ( X = aa_nat_nat(nth_nat(Xs),I) )
          & ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),I))
             => ~ pp(aa_nat_bool(P2,aa_nat_nat(nth_nat(Xs),J))) ) ) ) ).

% find_Some_iff
tff(fact_2380_find__Some__iff,axiom,
    ! [P2: fun_int_bool,Xs: list_int,X: int] :
      ( ( find_int(P2,Xs) = some_int(X) )
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
          & pp(aa_int_bool(P2,aa_nat_int(nth_int(Xs),I)))
          & ( X = aa_nat_int(nth_int(Xs),I) )
          & ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),I))
             => ~ pp(aa_int_bool(P2,aa_nat_int(nth_int(Xs),J))) ) ) ) ).

% find_Some_iff
tff(fact_2381_nth__Cons__pos,axiom,
    ! [N: nat,X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( nth_VEBT_VEBT(cons_VEBT_VEBT(X,Xs),N) = nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_2382_nth__Cons__pos,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_2383_rotate1__length01,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),one_one_nat))
     => ( rotate1_VEBT_VEBT(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_2384_rotate1__length01,axiom,
    ! [Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),one_one_nat))
     => ( rotate1_bool(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_2385_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat))
     => ( rotate1_nat(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_2386_rotate1__length01,axiom,
    ! [Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat))
     => ( rotate1_int(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_2387_length__rotate1,axiom,
    ! [Xs: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(rotate1_VEBT_VEBT(Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_rotate1
tff(fact_2388_length__rotate1,axiom,
    ! [Xs: list_bool] : size_size_list_bool(rotate1_bool(Xs)) = size_size_list_bool(Xs) ).

% length_rotate1
tff(fact_2389_length__rotate1,axiom,
    ! [Xs: list_nat] : aa_list_nat_nat(size_size_list_nat,rotate1_nat(Xs)) = aa_list_nat_nat(size_size_list_nat,Xs) ).

% length_rotate1
tff(fact_2390_length__rotate1,axiom,
    ! [Xs: list_int] : size_size_list_int(rotate1_int(Xs)) = size_size_list_int(Xs) ).

% length_rotate1
tff(fact_2391_nth__Cons__Suc,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] : nth_VEBT_VEBT(cons_VEBT_VEBT(X,Xs),aa_nat_nat(suc,N)) = nth_VEBT_VEBT(Xs,N) ).

% nth_Cons_Suc
tff(fact_2392_nth__Cons__Suc,axiom,
    ! [X: nat,Xs: list_nat,N: nat] : aa_nat_nat(nth_nat(cons_nat(X,Xs)),aa_nat_nat(suc,N)) = aa_nat_nat(nth_nat(Xs),N) ).

% nth_Cons_Suc
tff(fact_2393_nth__Cons__0,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] : nth_VEBT_VEBT(cons_VEBT_VEBT(X,Xs),zero_zero_nat) = X ).

% nth_Cons_0
tff(fact_2394_nth__Cons__0,axiom,
    ! [X: nat,Xs: list_nat] : aa_nat_nat(nth_nat(cons_nat(X,Xs)),zero_zero_nat) = X ).

% nth_Cons_0
tff(fact_2395_zip__Cons__Cons,axiom,
    ! [X: code_integer,Xs: list_Code_integer,Y: code_integer,Ys: list_Code_integer] : zip_Co3543743374963494515nteger(cons_Code_integer(X,Xs),cons_Code_integer(Y,Ys)) = cons_P9044669534377732177nteger(produc1086072967326762835nteger(X,Y),zip_Co3543743374963494515nteger(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_2396_zip__Cons__Cons,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] : zip_Pr4664179122662387191at_nat(cons_P6512896166579812791at_nat(X,Xs),cons_P6512896166579812791at_nat(Y,Ys)) = cons_P8732206157123786781at_nat(produc6161850002892822231at_nat(X,Y),zip_Pr4664179122662387191at_nat(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_2397_zip__Cons__Cons,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat] : zip_se5600341670672612855at_nat(cons_s6881495754146722583at_nat(X,Xs),cons_s6881495754146722583at_nat(Y,Ys)) = cons_P3940603068885512221at_nat(produc2922128104949294807at_nat(X,Y),zip_se5600341670672612855at_nat(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_2398_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] : zip_nat_nat(cons_nat(X,Xs),cons_nat(Y,Ys)) = cons_P6512896166579812791at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),zip_nat_nat(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_2399_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int] : zip_int_int(cons_int(X,Xs),cons_int(Y,Ys)) = cons_P3334398858971670639nt_int(product_Pair_int_int(X,Y),zip_int_int(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_2400_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: nat,Xs: list_nat] : enumerate_nat(N,cons_nat(X,Xs)) = cons_P6512896166579812791at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),X),enumerate_nat(aa_nat_nat(suc,N),Xs)) ).

% enumerate_simps(2)
tff(fact_2401_zip__eq__ConsE,axiom,
    ! [Xs: list_Code_integer,Ys: list_Code_integer,Xy: produc8923325533196201883nteger,Xys: list_P5578671422887162913nteger] :
      ( ( zip_Co3543743374963494515nteger(Xs,Ys) = cons_P9044669534377732177nteger(Xy,Xys) )
     => ~ ! [X3: code_integer,Xs4: list_Code_integer] :
            ( ( Xs = cons_Code_integer(X3,Xs4) )
           => ! [Y3: code_integer,Ys3: list_Code_integer] :
                ( ( Ys = cons_Code_integer(Y3,Ys3) )
               => ( ( Xy = produc1086072967326762835nteger(X3,Y3) )
                 => ( Xys != zip_Co3543743374963494515nteger(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_2402_zip__eq__ConsE,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Xy: produc859450856879609959at_nat,Xys: list_P8469869581646625389at_nat] :
      ( ( zip_Pr4664179122662387191at_nat(Xs,Ys) = cons_P8732206157123786781at_nat(Xy,Xys) )
     => ~ ! [X3: product_prod_nat_nat,Xs4: list_P6011104703257516679at_nat] :
            ( ( Xs = cons_P6512896166579812791at_nat(X3,Xs4) )
           => ! [Y3: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
                ( ( Ys = cons_P6512896166579812791at_nat(Y3,Ys3) )
               => ( ( Xy = produc6161850002892822231at_nat(X3,Y3) )
                 => ( Xys != zip_Pr4664179122662387191at_nat(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_2403_zip__eq__ConsE,axiom,
    ! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,Xy: produc3843707927480180839at_nat,Xys: list_P5464809261938338413at_nat] :
      ( ( zip_se5600341670672612855at_nat(Xs,Ys) = cons_P3940603068885512221at_nat(Xy,Xys) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat,Xs4: list_s1210847774152347623at_nat] :
            ( ( Xs = cons_s6881495754146722583at_nat(X3,Xs4) )
           => ! [Y3: set_Pr1261947904930325089at_nat,Ys3: list_s1210847774152347623at_nat] :
                ( ( Ys = cons_s6881495754146722583at_nat(Y3,Ys3) )
               => ( ( Xy = produc2922128104949294807at_nat(X3,Y3) )
                 => ( Xys != zip_se5600341670672612855at_nat(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_2404_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
      ( ( zip_nat_nat(Xs,Ys) = cons_P6512896166579812791at_nat(Xy,Xys) )
     => ~ ! [X3: nat,Xs4: list_nat] :
            ( ( Xs = cons_nat(X3,Xs4) )
           => ! [Y3: nat,Ys3: list_nat] :
                ( ( Ys = cons_nat(Y3,Ys3) )
               => ( ( Xy = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3) )
                 => ( Xys != zip_nat_nat(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_2405_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
      ( ( zip_int_int(Xs,Ys) = cons_P3334398858971670639nt_int(Xy,Xys) )
     => ~ ! [X3: int,Xs4: list_int] :
            ( ( Xs = cons_int(X3,Xs4) )
           => ! [Y3: int,Ys3: list_int] :
                ( ( Ys = cons_int(Y3,Ys3) )
               => ( ( Xy = product_Pair_int_int(X3,Y3) )
                 => ( Xys != zip_int_int(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_2406_find_Osimps_I2_J,axiom,
    ! [P2: fun_Pr7757163693017731481t_bool,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( pp(aa_Pro2232559814647311250t_bool(P2,X))
       => ( find_P8199882355184865565at_nat(P2,cons_P6512896166579812791at_nat(X,Xs)) = some_P7363390416028606310at_nat(X) ) )
      & ( ~ pp(aa_Pro2232559814647311250t_bool(P2,X))
       => ( find_P8199882355184865565at_nat(P2,cons_P6512896166579812791at_nat(X,Xs)) = find_P8199882355184865565at_nat(P2,Xs) ) ) ) ).

% find.simps(2)
tff(fact_2407_find_Osimps_I2_J,axiom,
    ! [P2: fun_nat_bool,X: nat,Xs: list_nat] :
      ( ( pp(aa_nat_bool(P2,X))
       => ( find_nat(P2,cons_nat(X,Xs)) = some_nat(X) ) )
      & ( ~ pp(aa_nat_bool(P2,X))
       => ( find_nat(P2,cons_nat(X,Xs)) = find_nat(P2,Xs) ) ) ) ).

% find.simps(2)
tff(fact_2408_find_Osimps_I2_J,axiom,
    ! [P2: fun_num_bool,X: num,Xs: list_num] :
      ( ( pp(aa_num_bool(P2,X))
       => ( find_num(P2,cons_num(X,Xs)) = some_num(X) ) )
      & ( ~ pp(aa_num_bool(P2,X))
       => ( find_num(P2,cons_num(X,Xs)) = find_num(P2,Xs) ) ) ) ).

% find.simps(2)
tff(fact_2409_set__zip__rightD,axiom,
    ! [X: code_integer,Y: code_integer,Xs: list_Code_integer,Ys: list_Code_integer] :
      ( member157494554546826820nteger(produc1086072967326762835nteger(X,Y),set_Pr920681315882439344nteger(zip_Co3543743374963494515nteger(Xs,Ys)))
     => member_Code_integer(Y,set_Code_integer2(Ys)) ) ).

% set_zip_rightD
tff(fact_2410_set__zip__rightD,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),set_Pr5518436109238095868at_nat(zip_Pr4664179122662387191at_nat(Xs,Ys)))
     => member8440522571783428010at_nat(Y,set_Pr5648618587558075414at_nat(Ys)) ) ).

% set_zip_rightD
tff(fact_2411_set__zip__rightD,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),set_Pr3765526544606949372at_nat(zip_se5600341670672612855at_nat(Xs,Ys)))
     => member2643936169264416010at_nat(Y,set_se5049602875457034614at_nat(Ys)) ) ).

% set_zip_rightD
tff(fact_2412_set__zip__rightD,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Ys)))
     => pp(aa_set_nat_bool(member_nat(Y),set_nat2(Ys))) ) ).

% set_zip_rightD
tff(fact_2413_set__zip__rightD,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(X,Y),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys)))
     => pp(aa_set_int_bool(member_int(Y),set_int2(Ys))) ) ).

% set_zip_rightD
tff(fact_2414_set__zip__leftD,axiom,
    ! [X: code_integer,Y: code_integer,Xs: list_Code_integer,Ys: list_Code_integer] :
      ( member157494554546826820nteger(produc1086072967326762835nteger(X,Y),set_Pr920681315882439344nteger(zip_Co3543743374963494515nteger(Xs,Ys)))
     => member_Code_integer(X,set_Code_integer2(Xs)) ) ).

% set_zip_leftD
tff(fact_2415_set__zip__leftD,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),set_Pr5518436109238095868at_nat(zip_Pr4664179122662387191at_nat(Xs,Ys)))
     => member8440522571783428010at_nat(X,set_Pr5648618587558075414at_nat(Xs)) ) ).

% set_zip_leftD
tff(fact_2416_set__zip__leftD,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),set_Pr3765526544606949372at_nat(zip_se5600341670672612855at_nat(Xs,Ys)))
     => member2643936169264416010at_nat(X,set_se5049602875457034614at_nat(Xs)) ) ).

% set_zip_leftD
tff(fact_2417_set__zip__leftD,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Ys)))
     => pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs))) ) ).

% set_zip_leftD
tff(fact_2418_set__zip__leftD,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(X,Y),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys)))
     => pp(aa_set_int_bool(member_int(X),set_int2(Xs))) ) ).

% set_zip_leftD
tff(fact_2419_in__set__zipE,axiom,
    ! [X: complex,Y: complex,Xs: list_complex,Ys: list_complex] :
      ( member5793383173714906214omplex(produc101793102246108661omplex(X,Y),set_Pr8199049879907524818omplex(zip_complex_complex(Xs,Ys)))
     => ~ ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
         => ~ pp(aa_set_complex_bool(member_complex(Y),set_complex2(Ys))) ) ) ).

% in_set_zipE
tff(fact_2420_in__set__zipE,axiom,
    ! [X: complex,Y: real,Xs: list_complex,Ys: list_real] :
      ( member47443559803733732x_real(produc1746590499379883635x_real(X,Y),set_Pr1225976482156248400x_real(zip_complex_real(Xs,Ys)))
     => ~ ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
         => ~ pp(aa_set_real_bool(member_real(Y),set_real2(Ys))) ) ) ).

% in_set_zipE
tff(fact_2421_in__set__zipE,axiom,
    ! [X: complex,Y: int,Xs: list_complex,Ys: list_int] :
      ( member595073364599660772ex_int(produc1367138851071493491ex_int(X,Y),set_Pr4995810437751016784ex_int(zip_complex_int(Xs,Ys)))
     => ~ ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
         => ~ pp(aa_set_int_bool(member_int(Y),set_int2(Ys))) ) ) ).

% in_set_zipE
tff(fact_2422_in__set__zipE,axiom,
    ! [X: real,Y: complex,Xs: list_real,Ys: list_complex] :
      ( member7358116576843751780omplex(produc1693001998875562995omplex(X,Y),set_Pr8536649499196266448omplex(zip_real_complex(Xs,Ys)))
     => ~ ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
         => ~ pp(aa_set_complex_bool(member_complex(Y),set_complex2(Ys))) ) ) ).

% in_set_zipE
tff(fact_2423_in__set__zipE,axiom,
    ! [X: real,Y: real,Xs: list_real,Ys: list_real] :
      ( member7849222048561428706l_real(produc4511245868158468465l_real(X,Y),set_Pr5999470521830281550l_real(zip_real_real(Xs,Ys)))
     => ~ ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
         => ~ pp(aa_set_real_bool(member_real(Y),set_real2(Ys))) ) ) ).

% in_set_zipE
tff(fact_2424_in__set__zipE,axiom,
    ! [X: real,Y: int,Xs: list_real,Ys: list_int] :
      ( member1627681773268152802al_int(produc3179012173361985393al_int(X,Y),set_Pr8219819362198175822al_int(zip_real_int(Xs,Ys)))
     => ~ ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
         => ~ pp(aa_set_int_bool(member_int(Y),set_int2(Ys))) ) ) ).

% in_set_zipE
tff(fact_2425_in__set__zipE,axiom,
    ! [X: int,Y: complex,Xs: list_int,Ys: list_complex] :
      ( member8811922270175639012omplex(produc7948753499206759283omplex(X,Y),set_Pr3989287306472219216omplex(zip_int_complex(Xs,Ys)))
     => ~ ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
         => ~ pp(aa_set_complex_bool(member_complex(Y),set_complex2(Ys))) ) ) ).

% in_set_zipE
tff(fact_2426_in__set__zipE,axiom,
    ! [X: int,Y: real,Xs: list_int,Ys: list_real] :
      ( member2744130022092475746t_real(produc801115645435158769t_real(X,Y),set_Pr112895574167722958t_real(zip_int_real(Xs,Ys)))
     => ~ ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
         => ~ pp(aa_set_real_bool(member_real(Y),set_real2(Ys))) ) ) ).

% in_set_zipE
tff(fact_2427_in__set__zipE,axiom,
    ! [X: complex,Y: vEBT_VEBT,Xs: list_complex,Ys: list_VEBT_VEBT] :
      ( member1978952105866562066T_VEBT(produc2757191886755552429T_VEBT(X,Y),set_Pr5158653123227461798T_VEBT(zip_co9157518722488180109T_VEBT(Xs,Ys)))
     => ~ ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
         => ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y),set_VEBT_VEBT2(Ys))) ) ) ).

% in_set_zipE
tff(fact_2428_in__set__zipE,axiom,
    ! [X: real,Y: vEBT_VEBT,Xs: list_real,Ys: list_VEBT_VEBT] :
      ( member7262085504369356948T_VEBT(produc6931449550656315951T_VEBT(X,Y),set_Pr8897343066327330088T_VEBT(zip_real_VEBT_VEBT(Xs,Ys)))
     => ~ ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
         => ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y),set_VEBT_VEBT2(Ys))) ) ) ).

% in_set_zipE
tff(fact_2429_zip__same,axiom,
    ! [A: complex,B: complex,Xs: list_complex] :
      ( member5793383173714906214omplex(produc101793102246108661omplex(A,B),set_Pr8199049879907524818omplex(zip_complex_complex(Xs,Xs)))
    <=> ( pp(aa_set_complex_bool(member_complex(A),set_complex2(Xs)))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_2430_zip__same,axiom,
    ! [A: real,B: real,Xs: list_real] :
      ( member7849222048561428706l_real(produc4511245868158468465l_real(A,B),set_Pr5999470521830281550l_real(zip_real_real(Xs,Xs)))
    <=> ( pp(aa_set_real_bool(member_real(A),set_real2(Xs)))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_2431_zip__same,axiom,
    ! [A: set_nat,B: set_nat,Xs: list_set_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(A,B),set_Pr9040384385603167362et_nat(zip_set_nat_set_nat(Xs,Xs)))
    <=> ( member_set_nat(A,set_set_nat2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_2432_zip__same,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member568628332442017744T_VEBT(produc537772716801021591T_VEBT(A,B),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Xs)))
    <=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),set_VEBT_VEBT2(Xs)))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_2433_zip__same,axiom,
    ! [A: code_integer,B: code_integer,Xs: list_Code_integer] :
      ( member157494554546826820nteger(produc1086072967326762835nteger(A,B),set_Pr920681315882439344nteger(zip_Co3543743374963494515nteger(Xs,Xs)))
    <=> ( member_Code_integer(A,set_Code_integer2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_2434_zip__same,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(A,B),set_Pr5518436109238095868at_nat(zip_Pr4664179122662387191at_nat(Xs,Xs)))
    <=> ( member8440522571783428010at_nat(A,set_Pr5648618587558075414at_nat(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_2435_zip__same,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A,B),set_Pr3765526544606949372at_nat(zip_se5600341670672612855at_nat(Xs,Xs)))
    <=> ( member2643936169264416010at_nat(A,set_se5049602875457034614at_nat(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_2436_zip__same,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Xs)))
    <=> ( pp(aa_set_nat_bool(member_nat(A),set_nat2(Xs)))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_2437_zip__same,axiom,
    ! [A: int,B: int,Xs: list_int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(A,B),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Xs)))
    <=> ( pp(aa_set_int_bool(member_int(A),set_int2(Xs)))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_2438_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( aa_nat_nat(suc,N) = size_s6755466524823107622T_VEBT(Xs) )
    <=> ? [Y5: vEBT_VEBT,Ys4: list_VEBT_VEBT] :
          ( ( Xs = cons_VEBT_VEBT(Y5,Ys4) )
          & ( size_s6755466524823107622T_VEBT(Ys4) = N ) ) ) ).

% Suc_length_conv
tff(fact_2439_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_bool] :
      ( ( aa_nat_nat(suc,N) = size_size_list_bool(Xs) )
    <=> ? [Y5: bool,Ys4: list_bool] :
          ( ( Xs = cons_bool(Y5,Ys4) )
          & ( size_size_list_bool(Ys4) = N ) ) ) ).

% Suc_length_conv
tff(fact_2440_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( aa_nat_nat(suc,N) = aa_list_nat_nat(size_size_list_nat,Xs) )
    <=> ? [Y5: nat,Ys4: list_nat] :
          ( ( Xs = cons_nat(Y5,Ys4) )
          & ( aa_list_nat_nat(size_size_list_nat,Ys4) = N ) ) ) ).

% Suc_length_conv
tff(fact_2441_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( aa_nat_nat(suc,N) = size_size_list_int(Xs) )
    <=> ? [Y5: int,Ys4: list_int] :
          ( ( Xs = cons_int(Y5,Ys4) )
          & ( size_size_list_int(Ys4) = N ) ) ) ).

% Suc_length_conv
tff(fact_2442_length__Suc__conv,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y5: vEBT_VEBT,Ys4: list_VEBT_VEBT] :
          ( ( Xs = cons_VEBT_VEBT(Y5,Ys4) )
          & ( size_s6755466524823107622T_VEBT(Ys4) = N ) ) ) ).

% length_Suc_conv
tff(fact_2443_length__Suc__conv,axiom,
    ! [Xs: list_bool,N: nat] :
      ( ( size_size_list_bool(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y5: bool,Ys4: list_bool] :
          ( ( Xs = cons_bool(Y5,Ys4) )
          & ( size_size_list_bool(Ys4) = N ) ) ) ).

% length_Suc_conv
tff(fact_2444_length__Suc__conv,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y5: nat,Ys4: list_nat] :
          ( ( Xs = cons_nat(Y5,Ys4) )
          & ( aa_list_nat_nat(size_size_list_nat,Ys4) = N ) ) ) ).

% length_Suc_conv
tff(fact_2445_length__Suc__conv,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( size_size_list_int(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y5: int,Ys4: list_int] :
          ( ( Xs = cons_int(Y5,Ys4) )
          & ( size_size_list_int(Ys4) = N ) ) ) ).

% length_Suc_conv
tff(fact_2446_impossible__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( Xs != cons_VEBT_VEBT(X,Ys) ) ) ).

% impossible_Cons
tff(fact_2447_impossible__Cons,axiom,
    ! [Xs: list_bool,Ys: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),size_size_list_bool(Ys)))
     => ( Xs != cons_bool(X,Ys) ) ) ).

% impossible_Cons
tff(fact_2448_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_nat_nat(size_size_list_nat,Ys)))
     => ( Xs != cons_nat(X,Ys) ) ) ).

% impossible_Cons
tff(fact_2449_impossible__Cons,axiom,
    ! [Xs: list_int,Ys: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),size_size_list_int(Ys)))
     => ( Xs != cons_int(X,Ys) ) ) ).

% impossible_Cons
tff(fact_2450_vebt__insert_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info2,zero_zero_nat,Ts2,S2),X) = vEBT_Node(Info2,zero_zero_nat,Ts2,S2) ).

% vebt_insert.simps(2)
tff(fact_2451_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_Code_integer,Ys: list_Code_integer,X: code_integer] :
      ( ( size_s3445333598471063425nteger(Xs) = size_s3445333598471063425nteger(Ys) )
     => ( member_Code_integer(X,set_Code_integer2(Xs))
       => ~ ! [Y3: code_integer] : ~ member157494554546826820nteger(produc1086072967326762835nteger(X,Y3),set_Pr920681315882439344nteger(zip_Co3543743374963494515nteger(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2452_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_complex,Ys: list_VEBT_VEBT,X: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
       => ~ ! [Y3: vEBT_VEBT] : ~ member1978952105866562066T_VEBT(produc2757191886755552429T_VEBT(X,Y3),set_Pr5158653123227461798T_VEBT(zip_co9157518722488180109T_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2453_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_VEBT_VEBT,X: real] :
      ( ( size_size_list_real(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
       => ~ ! [Y3: vEBT_VEBT] : ~ member7262085504369356948T_VEBT(produc6931449550656315951T_VEBT(X,Y3),set_Pr8897343066327330088T_VEBT(zip_real_VEBT_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2454_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_complex,Ys: list_bool,X: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = size_size_list_bool(Ys) )
     => ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
       => ~ ! [Y3: bool] : ~ member1660496674606185077x_bool(produc7867974272183215760x_bool(X,Y3),set_Pr3290376812781201673x_bool(zip_complex_bool(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2455_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_bool,X: real] :
      ( ( size_size_list_real(Xs) = size_size_list_bool(Ys) )
     => ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
       => ~ ! [Y3: bool] : ~ member3295258257541327095l_bool(produc3350604258665657362l_bool(X,Y3),set_Pr3131006541957657995l_bool(zip_real_bool(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2456_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_complex,Ys: list_nat,X: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
       => ~ ! [Y3: nat] : ~ member4772924384108857480ex_nat(produc1369629321580543767ex_nat(X,Y3),set_Pr9173661457260213492ex_nat(zip_complex_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2457_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_nat,X: real] :
      ( ( size_size_list_real(Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
       => ~ ! [Y3: nat] : ~ member5805532792777349510al_nat(produc3181502643871035669al_nat(X,Y3),set_Pr3174298344852596722al_nat(zip_real_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2458_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_complex,Ys: list_int,X: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = size_size_list_int(Ys) )
     => ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
       => ~ ! [Y3: int] : ~ member595073364599660772ex_int(produc1367138851071493491ex_int(X,Y3),set_Pr4995810437751016784ex_int(zip_complex_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2459_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_int,X: real] :
      ( ( size_size_list_real(Xs) = size_size_list_int(Ys) )
     => ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
       => ~ ! [Y3: int] : ~ member1627681773268152802al_int(produc3179012173361985393al_int(X,Y3),set_Pr8219819362198175822al_int(zip_real_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2460_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
       => ~ ! [Y3: vEBT_VEBT] : ~ member568628332442017744T_VEBT(produc537772716801021591T_VEBT(X,Y3),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2461_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_Code_integer,Ys: list_Code_integer,Y: code_integer] :
      ( ( size_s3445333598471063425nteger(Xs) = size_s3445333598471063425nteger(Ys) )
     => ( member_Code_integer(Y,set_Code_integer2(Ys))
       => ~ ! [X3: code_integer] : ~ member157494554546826820nteger(produc1086072967326762835nteger(X3,Y),set_Pr920681315882439344nteger(zip_Co3543743374963494515nteger(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2462_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_complex,Y: complex] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s3451745648224563538omplex(Ys) )
     => ( pp(aa_set_complex_bool(member_complex(Y),set_complex2(Ys)))
       => ~ ! [X3: vEBT_VEBT] : ~ member3207599676835851048omplex(produc5617778602380981643omplex(X3,Y),set_Pr6387300694196750780omplex(zip_VE2794733401258833515omplex(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2463_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_real,Y: real] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) )
     => ( pp(aa_set_real_bool(member_real(Y),set_real2(Ys)))
       => ~ ! [X3: vEBT_VEBT] : ~ member8675245146396747942T_real(produc8117437818029410057T_real(X3,Y),set_Pr1087130671499945274T_real(zip_VEBT_VEBT_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2464_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y),set_VEBT_VEBT2(Ys)))
       => ~ ! [X3: vEBT_VEBT] : ~ member568628332442017744T_VEBT(produc537772716801021591T_VEBT(X3,Y),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2465_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_bool,Y: bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Ys) )
     => ( member_bool(Y,set_bool2(Ys))
       => ~ ! [X3: vEBT_VEBT] : ~ member3504487111852181555T_bool(produc3035007008238225530T_bool(X3,Y),set_Pr3399341499479959199T_bool(zip_VEBT_VEBT_bool(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2466_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,Y: nat] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( pp(aa_set_nat_bool(member_nat(Y),set_nat2(Ys)))
       => ~ ! [X3: vEBT_VEBT] : ~ member373505688050248522BT_nat(produc738532404422230701BT_nat(X3,Y),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2467_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,Y: int] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) )
     => ( pp(aa_set_int_bool(member_int(Y),set_int2(Ys)))
       => ~ ! [X3: vEBT_VEBT] : ~ member5419026705395827622BT_int(produc736041933913180425BT_int(X3,Y),set_Pr2853735649769556538BT_int(zip_VEBT_VEBT_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2468_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_bool,Ys: list_complex,Y: complex] :
      ( ( size_size_list_bool(Xs) = size_s3451745648224563538omplex(Ys) )
     => ( pp(aa_set_complex_bool(member_complex(Y),set_complex2(Ys)))
       => ~ ! [X3: bool] : ~ member2723629924152719115omplex(produc5418556796337449326omplex(X3,Y),set_Pr4353510062327735711omplex(zip_bool_complex(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2469_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_bool,Ys: list_real,Y: real] :
      ( ( size_size_list_bool(Xs) = size_size_list_real(Ys) )
     => ( pp(aa_set_real_bool(member_real(Y),set_real2(Ys)))
       => ~ ! [X3: bool] : ~ member3680586897080482057l_real(produc7587643690356850540l_real(X3,Y),set_Pr3516335181496812957l_real(zip_bool_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2470_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_bool,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( size_size_list_bool(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y),set_VEBT_VEBT2(Ys)))
       => ~ ! [X3: bool] : ~ member5465154751058088621T_VEBT(produc9074556814250467700T_VEBT(X3,Y),set_Pr5360009138685866265T_VEBT(zip_bool_VEBT_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2471_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),size_s6755466524823107622T_VEBT(Xs)))
    <=> ? [X4: vEBT_VEBT,Ys4: list_VEBT_VEBT] :
          ( ( Xs = cons_VEBT_VEBT(X4,Ys4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s6755466524823107622T_VEBT(Ys4))) ) ) ).

% Suc_le_length_iff
tff(fact_2472_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),size_size_list_bool(Xs)))
    <=> ? [X4: bool,Ys4: list_bool] :
          ( ( Xs = cons_bool(X4,Ys4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_bool(Ys4))) ) ) ).

% Suc_le_length_iff
tff(fact_2473_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_list_nat_nat(size_size_list_nat,Xs)))
    <=> ? [X4: nat,Ys4: list_nat] :
          ( ( Xs = cons_nat(X4,Ys4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_nat_nat(size_size_list_nat,Ys4))) ) ) ).

% Suc_le_length_iff
tff(fact_2474_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),size_size_list_int(Xs)))
    <=> ? [X4: int,Ys4: list_int] :
          ( ( Xs = cons_int(X4,Ys4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_int(Ys4))) ) ) ).

% Suc_le_length_iff
tff(fact_2475_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X: nat] : vEBT_VEBT_insert(vEBT_Leaf(A,B),X) = vEBT_vebt_insert(vEBT_Leaf(A,B),X) ).

% VEBT_internal.insert'.simps(1)
tff(fact_2476_vebt__insert_Osimps_I3_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S2),X) = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) ).

% vebt_insert.simps(3)
tff(fact_2477_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_2478_list_Osize_I4_J,axiom,
    ! [X21: vEBT_VEBT,X22: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(cons_VEBT_VEBT(X21,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_s6755466524823107622T_VEBT(X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_2479_list_Osize_I4_J,axiom,
    ! [X21: bool,X22: list_bool] : size_size_list_bool(cons_bool(X21,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_bool(X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_2480_list_Osize_I4_J,axiom,
    ! [X21: nat,X22: list_nat] : aa_list_nat_nat(size_size_list_nat,cons_nat(X21,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_nat_nat(size_size_list_nat,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_2481_list_Osize_I4_J,axiom,
    ! [X21: int,X22: list_int] : size_size_list_int(cons_int(X21,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_int(X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_2482_nth__Cons_H,axiom,
    ! [N: nat,X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( nth_VEBT_VEBT(cons_VEBT_VEBT(X,Xs),N) = X ) )
      & ( ( N != zero_zero_nat )
       => ( nth_VEBT_VEBT(cons_VEBT_VEBT(X,Xs),N) = nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% nth_Cons'
tff(fact_2483_nth__Cons_H,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = X ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% nth_Cons'
tff(fact_2484_nth__equal__first__eq,axiom,
    ! [X: complex,Xs: list_complex,N: nat] :
      ( ~ pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s3451745648224563538omplex(Xs)))
       => ( ( aa_nat_complex(nth_complex(cons_complex(X,Xs)),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_2485_nth__equal__first__eq,axiom,
    ! [X: real,Xs: list_real,N: nat] :
      ( ~ pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_real(Xs)))
       => ( ( aa_nat_real(nth_real(cons_real(X,Xs)),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_2486_nth__equal__first__eq,axiom,
    ! [X: set_nat,Xs: list_set_nat,N: nat] :
      ( ~ member_set_nat(X,set_set_nat2(Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s3254054031482475050et_nat(Xs)))
       => ( ( aa_nat_set_nat(nth_set_nat(cons_set_nat(X,Xs)),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_2487_nth__equal__first__eq,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
      ( ~ 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_eq_nat,N),size_s6755466524823107622T_VEBT(Xs)))
       => ( ( nth_VEBT_VEBT(cons_VEBT_VEBT(X,Xs),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_2488_nth__equal__first__eq,axiom,
    ! [X: bool,Xs: list_bool,N: nat] :
      ( ~ member_bool(X,set_bool2(Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_bool(Xs)))
       => ( ( pp(aa_nat_bool(nth_bool(cons_bool(X,Xs)),N))
          <=> pp(X) )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_2489_nth__equal__first__eq,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_2490_nth__equal__first__eq,axiom,
    ! [X: int,Xs: list_int,N: nat] :
      ( ~ pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_int(Xs)))
       => ( ( aa_nat_int(nth_int(cons_int(X,Xs)),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_2491_nth__non__equal__first__eq,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
      ( ( X != Y )
     => ( ( nth_VEBT_VEBT(cons_VEBT_VEBT(X,Xs),N) = Y )
      <=> ( ( nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_2492_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,N: nat] :
      ( ( X != Y )
     => ( ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = Y )
      <=> ( ( aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_2493_length__Cons,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(cons_VEBT_VEBT(X,Xs)) = aa_nat_nat(suc,size_s6755466524823107622T_VEBT(Xs)) ).

% length_Cons
tff(fact_2494_length__Cons,axiom,
    ! [X: bool,Xs: list_bool] : size_size_list_bool(cons_bool(X,Xs)) = aa_nat_nat(suc,size_size_list_bool(Xs)) ).

% length_Cons
tff(fact_2495_length__Cons,axiom,
    ! [X: nat,Xs: list_nat] : aa_list_nat_nat(size_size_list_nat,cons_nat(X,Xs)) = aa_nat_nat(suc,aa_list_nat_nat(size_size_list_nat,Xs)) ).

% length_Cons
tff(fact_2496_length__Cons,axiom,
    ! [X: int,Xs: list_int] : size_size_list_int(cons_int(X,Xs)) = aa_nat_nat(suc,size_size_list_int(Xs)) ).

% length_Cons
tff(fact_2497_dbl__inc__simps_I2_J,axiom,
    neg_nu5831290666863070958nteger(zero_z3403309356797280102nteger) = one_one_Code_integer ).

% dbl_inc_simps(2)
tff(fact_2498_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_2499_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_2500_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_2501_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_2502_enumerate__Suc_H,axiom,
    ! [S3: set_nat,N: nat] : aa_nat_nat(infini8530281810654367211te_nat(S3),aa_nat_nat(suc,N)) = aa_nat_nat(infini8530281810654367211te_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S3),insert_nat(aa_nat_nat(infini8530281810654367211te_nat(S3),zero_zero_nat),bot_bot_set_nat))),N) ).

% enumerate_Suc'
tff(fact_2503_dbl__dec__def,axiom,
    ! [X: code_integer] : neg_nu7757733837767384882nteger(X) = minus_8373710615458151222nteger(plus_p5714425477246183910nteger(X,X),one_one_Code_integer) ).

% dbl_dec_def
tff(fact_2504_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_2505_dbl__dec__def,axiom,
    ! [X: real] : aa_real_real(neg_nu6075765906172075777c_real,X) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(X),X)),one_one_real) ).

% dbl_dec_def
tff(fact_2506_dbl__dec__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,X) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X)),one_one_rat) ).

% dbl_dec_def
tff(fact_2507_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_2508_Gcd__fin__0__iff,axiom,
    ! [A4: set_int] :
      ( ( semiri4256215615220890538in_int(A4) = zero_zero_int )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),insert_int(zero_zero_int,bot_bot_set_int)))
        & pp(aa_set_int_bool(finite_finite_int,A4)) ) ) ).

% Gcd_fin_0_iff
tff(fact_2509_Gcd__fin__0__iff,axiom,
    ! [A4: set_nat] :
      ( ( aa_set_nat_nat(semiri4258706085729940814in_nat,A4) = zero_zero_nat )
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A4),insert_nat(zero_zero_nat,bot_bot_set_nat)))
        & pp(aa_set_nat_bool(finite_finite_nat,A4)) ) ) ).

% Gcd_fin_0_iff
tff(fact_2510_power__decreasing__iff,axiom,
    ! [B: code_integer,M2: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),one_one_Code_integer))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),M2)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ) ) ).

% power_decreasing_iff
tff(fact_2511_power__decreasing__iff,axiom,
    ! [B: real,M2: nat,N: 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(aa_real_fun_nat_real(power_power_real,B),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ) ) ).

% power_decreasing_iff
tff(fact_2512_power__decreasing__iff,axiom,
    ! [B: rat,M2: nat,N: 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),M2)),aa_nat_rat(power_power_rat(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ) ) ).

% power_decreasing_iff
tff(fact_2513_power__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: 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),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ) ) ).

% power_decreasing_iff
tff(fact_2514_power__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: 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),M2)),aa_nat_int(power_power_int(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ) ) ).

% power_decreasing_iff
tff(fact_2515_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_real,X: real] :
      ( 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(ord_less_eq_nat,finite_card_real(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(insert_real(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_2516_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_nat,X: 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(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,insert_nat(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_2517_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_complex,X: complex] :
      ( 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(ord_less_eq_nat,finite_card_complex(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(insert_complex(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_2518_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_int,X: int] :
      ( 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(ord_less_eq_nat,finite_card_int(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(insert_int(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_2519_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_list_nat,X: list_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(ord_less_eq_nat,finite_card_list_nat(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(insert_list_nat(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_2520_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_set_nat,X: set_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(ord_less_eq_nat,finite_card_set_nat(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(insert_set_nat(X,Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_2521_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_2522_count__notin,axiom,
    ! [X: real,Xs: list_real] :
      ( ~ pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
     => ( aa_real_nat(count_list_real(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_2523_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_2524_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_2525_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_2526_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_2527_Cons__lenlex__iff,axiom,
    ! [M2: code_integer,Ms: list_Code_integer,N: code_integer,Ns: list_Code_integer,R: set_Pr4811707699266497531nteger] :
      ( member749217712838834276nteger(produc750622340256944499nteger(cons_Code_integer(M2,Ms),cons_Code_integer(N,Ns)),lenlex_Code_integer(R))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s3445333598471063425nteger(Ms)),size_s3445333598471063425nteger(Ns)))
        | ( ( size_s3445333598471063425nteger(Ms) = size_s3445333598471063425nteger(Ns) )
          & member157494554546826820nteger(produc1086072967326762835nteger(M2,N),R) )
        | ( ( M2 = N )
          & member749217712838834276nteger(produc750622340256944499nteger(Ms,Ns),lenlex_Code_integer(R)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_2528_Cons__lenlex__iff,axiom,
    ! [M2: product_prod_nat_nat,Ms: list_P6011104703257516679at_nat,N: product_prod_nat_nat,Ns: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
      ( member6693912407220327184at_nat(produc5943733680697469783at_nat(cons_P6512896166579812791at_nat(M2,Ms),cons_P6512896166579812791at_nat(N,Ns)),lenlex325483962726685836at_nat(R))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s5460976970255530739at_nat(Ms)),size_s5460976970255530739at_nat(Ns)))
        | ( ( size_s5460976970255530739at_nat(Ms) = size_s5460976970255530739at_nat(Ns) )
          & member8206827879206165904at_nat(produc6161850002892822231at_nat(M2,N),R) )
        | ( ( M2 = N )
          & member6693912407220327184at_nat(produc5943733680697469783at_nat(Ms,Ns),lenlex325483962726685836at_nat(R)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_2529_Cons__lenlex__iff,axiom,
    ! [M2: set_Pr1261947904930325089at_nat,Ms: list_s1210847774152347623at_nat,N: set_Pr1261947904930325089at_nat,Ns: list_s1210847774152347623at_nat,R: set_Pr4329608150637261639at_nat] :
      ( member4080735728053443344at_nat(produc7536900900485677911at_nat(cons_s6881495754146722583at_nat(M2,Ms),cons_s6881495754146722583at_nat(N,Ns)),lenlex1357538814655152620at_nat(R))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s8736152011456118867at_nat(Ms)),size_s8736152011456118867at_nat(Ns)))
        | ( ( size_s8736152011456118867at_nat(Ms) = size_s8736152011456118867at_nat(Ns) )
          & member8757157785044589968at_nat(produc2922128104949294807at_nat(M2,N),R) )
        | ( ( M2 = N )
          & member4080735728053443344at_nat(produc7536900900485677911at_nat(Ms,Ns),lenlex1357538814655152620at_nat(R)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_2530_Cons__lenlex__iff,axiom,
    ! [M2: vEBT_VEBT,Ms: list_VEBT_VEBT,N: vEBT_VEBT,Ns: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(cons_VEBT_VEBT(M2,Ms),cons_VEBT_VEBT(N,Ns)),lenlex_VEBT_VEBT(R))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(Ms)),size_s6755466524823107622T_VEBT(Ns)))
        | ( ( size_s6755466524823107622T_VEBT(Ms) = size_s6755466524823107622T_VEBT(Ns) )
          & member568628332442017744T_VEBT(produc537772716801021591T_VEBT(M2,N),R) )
        | ( ( M2 = N )
          & member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Ms,Ns),lenlex_VEBT_VEBT(R)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_2531_Cons__lenlex__iff,axiom,
    ! [M2: bool,Ms: list_bool,N: bool,Ns: list_bool,R: set_Pr8621012493645324487l_bool] :
      ( member6300827366518646928t_bool(produc970142415094439895t_bool(cons_bool(M2,Ms),cons_bool(N,Ns)),lenlex_bool(R))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(Ms)),size_size_list_bool(Ns)))
        | ( ( size_size_list_bool(Ms) = size_size_list_bool(Ns) )
          & member4869097156692242192l_bool(produc8297744024737662807l_bool(M2,N),R) )
        | ( ( pp(M2)
          <=> pp(N) )
          & member6300827366518646928t_bool(produc970142415094439895t_bool(Ms,Ns),lenlex_bool(R)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_2532_Cons__lenlex__iff,axiom,
    ! [M2: nat,Ms: list_nat,N: nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(M2,Ms),cons_nat(N,Ns)),lenlex_nat(R))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_nat_nat(size_size_list_nat,Ms)),aa_list_nat_nat(size_size_list_nat,Ns)))
        | ( ( aa_list_nat_nat(size_size_list_nat,Ms) = aa_list_nat_nat(size_size_list_nat,Ns) )
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M2),N),R) )
        | ( ( M2 = N )
          & member7340969449405702474st_nat(produc2694037385005941721st_nat(Ms,Ns),lenlex_nat(R)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_2533_Cons__lenlex__iff,axiom,
    ! [M2: int,Ms: list_int,N: int,Ns: list_int,R: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(cons_int(M2,Ms),cons_int(N,Ns)),lenlex_int(R))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(Ms)),size_size_list_int(Ns)))
        | ( ( size_size_list_int(Ms) = size_size_list_int(Ns) )
          & member5262025264175285858nt_int(product_Pair_int_int(M2,N),R) )
        | ( ( M2 = N )
          & member6698963635872716290st_int(produc364263696895485585st_int(Ms,Ns),lenlex_int(R)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_2534_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_2535_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_2536_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M2 = zero_zero_nat )
        | ( X = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_2537_power__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_2538_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: 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),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_2539_dbl__dec__simps_I3_J,axiom,
    neg_nu7757733837767384882nteger(one_one_Code_integer) = one_one_Code_integer ).

% dbl_dec_simps(3)
tff(fact_2540_dbl__dec__simps_I3_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).

% dbl_dec_simps(3)
tff(fact_2541_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_2542_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_2543_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,N)) = zero_zero_rat ).

% power_0_Suc
tff(fact_2544_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,N)) = zero_zero_nat ).

% power_0_Suc
tff(fact_2545_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,N)) = zero_zero_real ).

% power_0_Suc
tff(fact_2546_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,N)) = zero_zero_int ).

% power_0_Suc
tff(fact_2547_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,N)) = zero_zero_complex ).

% power_0_Suc
tff(fact_2548_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_2549_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_2550_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_2551_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_2552_card_Oempty,axiom,
    finite_card_complex(bot_bot_set_complex) = zero_zero_nat ).

% card.empty
tff(fact_2553_card_Oempty,axiom,
    finite_card_list_nat(bot_bot_set_list_nat) = zero_zero_nat ).

% card.empty
tff(fact_2554_card_Oempty,axiom,
    finite_card_set_nat(bot_bot_set_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_2555_card_Oempty,axiom,
    aa_set_nat_nat(finite_card_nat,bot_bot_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_2556_card_Oempty,axiom,
    finite_card_int(bot_bot_set_int) = zero_zero_nat ).

% card.empty
tff(fact_2557_card_Oempty,axiom,
    finite_card_real(bot_bot_set_real) = zero_zero_nat ).

% card.empty
tff(fact_2558_card_Oinfinite,axiom,
    ! [A4: set_list_nat] :
      ( ~ finite8100373058378681591st_nat(A4)
     => ( finite_card_list_nat(A4) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_2559_card_Oinfinite,axiom,
    ! [A4: set_set_nat] :
      ( ~ finite1152437895449049373et_nat(A4)
     => ( finite_card_set_nat(A4) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_2560_card_Oinfinite,axiom,
    ! [A4: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( aa_set_nat_nat(finite_card_nat,A4) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_2561_card_Oinfinite,axiom,
    ! [A4: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
     => ( finite_card_int(A4) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_2562_card_Oinfinite,axiom,
    ! [A4: set_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( finite_card_complex(A4) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_2563_Gcd__fin_Oempty,axiom,
    aa_set_nat_nat(semiri4258706085729940814in_nat,bot_bot_set_nat) = zero_zero_nat ).

% Gcd_fin.empty
tff(fact_2564_Gcd__fin_Oempty,axiom,
    semiri4256215615220890538in_int(bot_bot_set_int) = zero_zero_int ).

% Gcd_fin.empty
tff(fact_2565_power__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(power_power_rat(A),N) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_2566_power__eq__0__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_2567_power__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_2568_power__eq__0__iff,axiom,
    ! [A: int,N: nat] :
      ( ( aa_nat_int(power_power_int(A),N) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_2569_power__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( aa_nat_complex(power_power_complex(A),N) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_2570_card__0__eq,axiom,
    ! [A4: set_list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( ( finite_card_list_nat(A4) = zero_zero_nat )
      <=> ( A4 = bot_bot_set_list_nat ) ) ) ).

% card_0_eq
tff(fact_2571_card__0__eq,axiom,
    ! [A4: set_set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( ( finite_card_set_nat(A4) = zero_zero_nat )
      <=> ( A4 = bot_bot_set_set_nat ) ) ) ).

% card_0_eq
tff(fact_2572_card__0__eq,axiom,
    ! [A4: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( finite_card_complex(A4) = zero_zero_nat )
      <=> ( A4 = bot_bot_set_complex ) ) ) ).

% card_0_eq
tff(fact_2573_card__0__eq,axiom,
    ! [A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( aa_set_nat_nat(finite_card_nat,A4) = zero_zero_nat )
      <=> ( A4 = bot_bot_set_nat ) ) ) ).

% card_0_eq
tff(fact_2574_card__0__eq,axiom,
    ! [A4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( finite_card_int(A4) = zero_zero_nat )
      <=> ( A4 = bot_bot_set_int ) ) ) ).

% card_0_eq
tff(fact_2575_card__0__eq,axiom,
    ! [A4: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( finite_card_real(A4) = zero_zero_nat )
      <=> ( A4 = bot_bot_set_real ) ) ) ).

% card_0_eq
tff(fact_2576_card__insert__disjoint,axiom,
    ! [A4: set_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ~ pp(aa_set_real_bool(member_real(X),A4))
       => ( finite_card_real(insert_real(X,A4)) = aa_nat_nat(suc,finite_card_real(A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_2577_card__insert__disjoint,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( ~ member_list_nat(X,A4)
       => ( finite_card_list_nat(insert_list_nat(X,A4)) = aa_nat_nat(suc,finite_card_list_nat(A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_2578_card__insert__disjoint,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( ~ member_set_nat(X,A4)
       => ( finite_card_set_nat(insert_set_nat(X,A4)) = aa_nat_nat(suc,finite_card_set_nat(A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_2579_card__insert__disjoint,axiom,
    ! [A4: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
       => ( aa_set_nat_nat(finite_card_nat,insert_nat(X,A4)) = aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_2580_card__insert__disjoint,axiom,
    ! [A4: set_int,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ~ pp(aa_set_int_bool(member_int(X),A4))
       => ( finite_card_int(insert_int(X,A4)) = aa_nat_nat(suc,finite_card_int(A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_2581_card__insert__disjoint,axiom,
    ! [A4: set_complex,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A4))
       => ( finite_card_complex(insert_complex(X,A4)) = aa_nat_nat(suc,finite_card_complex(A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_2582_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M2: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),one_one_Code_integer))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),M2)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2583_power__strict__decreasing__iff,axiom,
    ! [B: real,M2: nat,N: 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(aa_real_fun_nat_real(power_power_real,B),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2584_power__strict__decreasing__iff,axiom,
    ! [B: rat,M2: nat,N: 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),M2)),aa_nat_rat(power_power_rat(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2585_power__strict__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: 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),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2586_power__strict__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: 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),M2)),aa_nat_int(power_power_int(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2587_power__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),X)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2588_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(aa_real_fun_nat_real(power_power_real,B),X)),aa_nat_real(aa_real_fun_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_2589_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_2590_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_2591_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_2592_power__mono__iff,axiom,
    ! [A: real,B: real,N: 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),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2593_power__mono__iff,axiom,
    ! [A: rat,B: rat,N: 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),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)))
          <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2594_power__mono__iff,axiom,
    ! [A: nat,B: nat,N: 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),N))
         => ( 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),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2595_power__mono__iff,axiom,
    ! [A: int,B: int,N: 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),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
          <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2596_power__not__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),N) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_2597_power__not__zero,axiom,
    ! [A: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_2598_power__not__zero,axiom,
    ! [A: real,N: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_2599_power__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),N) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_2600_power__not__zero,axiom,
    ! [A: complex,N: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),N) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_2601_power__commuting__commutes,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( aa_complex_complex(times_times_complex(X),Y) = aa_complex_complex(times_times_complex(Y),X) )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),N)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(power_power_complex(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_2602_power__commuting__commutes,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) ) ) ).

% power_commuting_commutes
tff(fact_2603_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),N)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(power_power_rat(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_2604_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),X) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)) ) ) ).

% power_commuting_commutes
tff(fact_2605_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),X) )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),N)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(power_power_int(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_2606_power__mult__distrib,axiom,
    ! [A: complex,B: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(times_times_complex(A),B)),N) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),N)),aa_nat_complex(power_power_complex(B),N)) ).

% power_mult_distrib
tff(fact_2607_power__mult__distrib,axiom,
    ! [A: real,B: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)) ).

% power_mult_distrib
tff(fact_2608_power__mult__distrib,axiom,
    ! [A: rat,B: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)) ).

% power_mult_distrib
tff(fact_2609_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)) ).

% power_mult_distrib
tff(fact_2610_power__mult__distrib,axiom,
    ! [A: int,B: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)) ).

% power_mult_distrib
tff(fact_2611_power__commutes,axiom,
    ! [A: complex,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),N)),A) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),N)) ).

% power_commutes
tff(fact_2612_power__commutes,axiom,
    ! [A: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% power_commutes
tff(fact_2613_power__commutes,axiom,
    ! [A: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),N)),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N)) ).

% power_commutes
tff(fact_2614_power__commutes,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_commutes
tff(fact_2615_power__commutes,axiom,
    ! [A: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)) ).

% power_commutes
tff(fact_2616_power__mult,axiom,
    ! [A: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),N) ).

% power_mult
tff(fact_2617_power__mult,axiom,
    ! [A: real,M2: nat,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2)),N) ).

% power_mult
tff(fact_2618_power__mult,axiom,
    ! [A: int,M2: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),M2)),N) ).

% power_mult
tff(fact_2619_power__mult,axiom,
    ! [A: complex,M2: nat,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),M2)),N) ).

% power_mult
tff(fact_2620_finite__le__enumerate,axiom,
    ! [S3: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_set_nat_nat(finite_card_nat,S3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(infini8530281810654367211te_nat(S3),N))) ) ) ).

% finite_le_enumerate
tff(fact_2621_lenlex__irreflexive,axiom,
    ! [R: set_Pr4811707699266497531nteger,Xs: list_Code_integer] :
      ( ! [X3: code_integer] : ~ member157494554546826820nteger(produc1086072967326762835nteger(X3,X3),R)
     => ~ member749217712838834276nteger(produc750622340256944499nteger(Xs,Xs),lenlex_Code_integer(R)) ) ).

% lenlex_irreflexive
tff(fact_2622_lenlex__irreflexive,axiom,
    ! [R: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat] :
      ( ! [X3: product_prod_nat_nat] : ~ member8206827879206165904at_nat(produc6161850002892822231at_nat(X3,X3),R)
     => ~ member6693912407220327184at_nat(produc5943733680697469783at_nat(Xs,Xs),lenlex325483962726685836at_nat(R)) ) ).

% lenlex_irreflexive
tff(fact_2623_lenlex__irreflexive,axiom,
    ! [R: set_Pr4329608150637261639at_nat,Xs: list_s1210847774152347623at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat] : ~ member8757157785044589968at_nat(produc2922128104949294807at_nat(X3,X3),R)
     => ~ member4080735728053443344at_nat(produc7536900900485677911at_nat(Xs,Xs),lenlex1357538814655152620at_nat(R)) ) ).

% lenlex_irreflexive
tff(fact_2624_lenlex__irreflexive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ! [X3: nat] : ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),X3),R)
     => ~ member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Xs),lenlex_nat(R)) ) ).

% lenlex_irreflexive
tff(fact_2625_lenlex__irreflexive,axiom,
    ! [R: set_Pr958786334691620121nt_int,Xs: list_int] :
      ( ! [X3: int] : ~ member5262025264175285858nt_int(product_Pair_int_int(X3,X3),R)
     => ~ member6698963635872716290st_int(produc364263696895485585st_int(Xs,Xs),lenlex_int(R)) ) ).

% lenlex_irreflexive
tff(fact_2626_zero__le__power,axiom,
    ! [A: real,N: 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(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% zero_le_power
tff(fact_2627_zero__le__power,axiom,
    ! [A: rat,N: 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),N))) ) ).

% zero_le_power
tff(fact_2628_zero__le__power,axiom,
    ! [A: nat,N: 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),N))) ) ).

% zero_le_power
tff(fact_2629_zero__le__power,axiom,
    ! [A: int,N: 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),N))) ) ).

% zero_le_power
tff(fact_2630_power__mono,axiom,
    ! [A: real,B: real,N: 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(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ).

% power_mono
tff(fact_2631_power__mono,axiom,
    ! [A: rat,B: rat,N: 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),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).

% power_mono
tff(fact_2632_power__mono,axiom,
    ! [A: nat,B: nat,N: 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),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))) ) ) ).

% power_mono
tff(fact_2633_power__mono,axiom,
    ! [A: int,B: int,N: 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),N)),aa_nat_int(power_power_int(B),N))) ) ) ).

% power_mono
tff(fact_2634_finite__enumerate__step,axiom,
    ! [S3: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),aa_set_nat_nat(finite_card_nat,S3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S3),N)),aa_nat_nat(infini8530281810654367211te_nat(S3),aa_nat_nat(suc,N)))) ) ) ).

% finite_enumerate_step
tff(fact_2635_card__le__if__inj__on__rel,axiom,
    ! [B5: set_real,A4: set_real,R: fun_re1255401033344146673l_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,B5))
     => ( ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A4))
           => ? [B6: real] :
                ( pp(aa_set_real_bool(member_real(B6),B5))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,A3),B6)) ) )
       => ( ! [A1: real,A22: real,B3: real] :
              ( pp(aa_set_real_bool(member_real(A1),A4))
             => ( pp(aa_set_real_bool(member_real(A22),A4))
               => ( pp(aa_set_real_bool(member_real(B3),B5))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,A1),B3))
                   => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,A22),B3))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(A4)),finite_card_real(B5))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_2636_card__le__if__inj__on__rel,axiom,
    ! [B5: set_real,A4: set_nat,R: fun_na6801867027684862029l_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,B5))
     => ( ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A4))
           => ? [B6: real] :
                ( pp(aa_set_real_bool(member_real(B6),B5))
                & pp(aa_real_bool(aa_nat_fun_real_bool(R,A3),B6)) ) )
       => ( ! [A1: nat,A22: nat,B3: real] :
              ( pp(aa_set_nat_bool(member_nat(A1),A4))
             => ( pp(aa_set_nat_bool(member_nat(A22),A4))
               => ( pp(aa_set_real_bool(member_real(B3),B5))
                 => ( pp(aa_real_bool(aa_nat_fun_real_bool(R,A1),B3))
                   => ( pp(aa_real_bool(aa_nat_fun_real_bool(R,A22),B3))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),finite_card_real(B5))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_2637_card__le__if__inj__on__rel,axiom,
    ! [B5: set_real,A4: set_complex,R: fun_co2286642472385812463l_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,B5))
     => ( ! [A3: complex] :
            ( pp(aa_set_complex_bool(member_complex(A3),A4))
           => ? [B6: real] :
                ( pp(aa_set_real_bool(member_real(B6),B5))
                & pp(aa_real_bool(aa_com8943241093633526360l_bool(R,A3),B6)) ) )
       => ( ! [A1: complex,A22: complex,B3: real] :
              ( pp(aa_set_complex_bool(member_complex(A1),A4))
             => ( pp(aa_set_complex_bool(member_complex(A22),A4))
               => ( pp(aa_set_real_bool(member_real(B3),B5))
                 => ( pp(aa_real_bool(aa_com8943241093633526360l_bool(R,A1),B3))
                   => ( pp(aa_real_bool(aa_com8943241093633526360l_bool(R,A22),B3))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A4)),finite_card_real(B5))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_2638_card__le__if__inj__on__rel,axiom,
    ! [B5: set_real,A4: set_int,R: fun_in3970322703178350321l_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,B5))
     => ( ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A4))
           => ? [B6: real] :
                ( pp(aa_set_real_bool(member_real(B6),B5))
                & pp(aa_real_bool(aa_int_fun_real_bool(R,A3),B6)) ) )
       => ( ! [A1: int,A22: int,B3: real] :
              ( pp(aa_set_int_bool(member_int(A1),A4))
             => ( pp(aa_set_int_bool(member_int(A22),A4))
               => ( pp(aa_set_real_bool(member_real(B3),B5))
                 => ( pp(aa_real_bool(aa_int_fun_real_bool(R,A1),B3))
                   => ( pp(aa_real_bool(aa_int_fun_real_bool(R,A22),B3))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(A4)),finite_card_real(B5))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_2639_card__le__if__inj__on__rel,axiom,
    ! [B5: set_nat,A4: set_real,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B5))
     => ( ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A4))
           => ? [B6: nat] :
                ( pp(aa_set_nat_bool(member_nat(B6),B5))
                & pp(aa_nat_bool(aa_real_fun_nat_bool(R,A3),B6)) ) )
       => ( ! [A1: real,A22: real,B3: nat] :
              ( pp(aa_set_real_bool(member_real(A1),A4))
             => ( pp(aa_set_real_bool(member_real(A22),A4))
               => ( pp(aa_set_nat_bool(member_nat(B3),B5))
                 => ( pp(aa_nat_bool(aa_real_fun_nat_bool(R,A1),B3))
                   => ( pp(aa_nat_bool(aa_real_fun_nat_bool(R,A22),B3))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(A4)),aa_set_nat_nat(finite_card_nat,B5))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_2640_card__le__if__inj__on__rel,axiom,
    ! [B5: set_nat,A4: set_nat,R: fun_nat_fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B5))
     => ( ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A4))
           => ? [B6: nat] :
                ( pp(aa_set_nat_bool(member_nat(B6),B5))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,A3),B6)) ) )
       => ( ! [A1: nat,A22: nat,B3: nat] :
              ( pp(aa_set_nat_bool(member_nat(A1),A4))
             => ( pp(aa_set_nat_bool(member_nat(A22),A4))
               => ( pp(aa_set_nat_bool(member_nat(B3),B5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,A1),B3))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,A22),B3))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B5))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_2641_card__le__if__inj__on__rel,axiom,
    ! [B5: set_nat,A4: set_complex,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B5))
     => ( ! [A3: complex] :
            ( pp(aa_set_complex_bool(member_complex(A3),A4))
           => ? [B6: nat] :
                ( pp(aa_set_nat_bool(member_nat(B6),B5))
                & pp(aa_nat_bool(aa_com5335841084339320244t_bool(R,A3),B6)) ) )
       => ( ! [A1: complex,A22: complex,B3: nat] :
              ( pp(aa_set_complex_bool(member_complex(A1),A4))
             => ( pp(aa_set_complex_bool(member_complex(A22),A4))
               => ( pp(aa_set_nat_bool(member_nat(B3),B5))
                 => ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(R,A1),B3))
                   => ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(R,A22),B3))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A4)),aa_set_nat_nat(finite_card_nat,B5))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_2642_card__le__if__inj__on__rel,axiom,
    ! [B5: set_nat,A4: set_int,R: fun_int_fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B5))
     => ( ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A4))
           => ? [B6: nat] :
                ( pp(aa_set_nat_bool(member_nat(B6),B5))
                & pp(aa_nat_bool(aa_int_fun_nat_bool(R,A3),B6)) ) )
       => ( ! [A1: int,A22: int,B3: nat] :
              ( pp(aa_set_int_bool(member_int(A1),A4))
             => ( pp(aa_set_int_bool(member_int(A22),A4))
               => ( pp(aa_set_nat_bool(member_nat(B3),B5))
                 => ( pp(aa_nat_bool(aa_int_fun_nat_bool(R,A1),B3))
                   => ( pp(aa_nat_bool(aa_int_fun_nat_bool(R,A22),B3))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(A4)),aa_set_nat_nat(finite_card_nat,B5))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_2643_card__le__if__inj__on__rel,axiom,
    ! [B5: set_int,A4: set_real,R: fun_re8661203372455305201t_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,B5))
     => ( ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A4))
           => ? [B6: int] :
                ( pp(aa_set_int_bool(member_int(B6),B5))
                & pp(aa_int_bool(aa_real_fun_int_bool(R,A3),B6)) ) )
       => ( ! [A1: real,A22: real,B3: int] :
              ( pp(aa_set_real_bool(member_real(A1),A4))
             => ( pp(aa_set_real_bool(member_real(A22),A4))
               => ( pp(aa_set_int_bool(member_int(B3),B5))
                 => ( pp(aa_int_bool(aa_real_fun_int_bool(R,A1),B3))
                   => ( pp(aa_int_bool(aa_real_fun_int_bool(R,A22),B3))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(A4)),finite_card_int(B5))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_2644_card__le__if__inj__on__rel,axiom,
    ! [B5: set_int,A4: set_nat,R: fun_nat_fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,B5))
     => ( ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A4))
           => ? [B6: int] :
                ( pp(aa_set_int_bool(member_int(B6),B5))
                & pp(aa_int_bool(aa_nat_fun_int_bool(R,A3),B6)) ) )
       => ( ! [A1: nat,A22: nat,B3: int] :
              ( pp(aa_set_nat_bool(member_nat(A1),A4))
             => ( pp(aa_set_nat_bool(member_nat(A22),A4))
               => ( pp(aa_set_int_bool(member_int(B3),B5))
                 => ( pp(aa_int_bool(aa_nat_fun_int_bool(R,A1),B3))
                   => ( pp(aa_int_bool(aa_nat_fun_int_bool(R,A22),B3))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),finite_card_int(B5))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_2645_zero__less__power,axiom,
    ! [A: real,N: 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(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% zero_less_power
tff(fact_2646_zero__less__power,axiom,
    ! [A: rat,N: 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),N))) ) ).

% zero_less_power
tff(fact_2647_zero__less__power,axiom,
    ! [A: nat,N: 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),N))) ) ).

% zero_less_power
tff(fact_2648_zero__less__power,axiom,
    ! [A: int,N: 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),N))) ) ).

% zero_less_power
tff(fact_2649_card__insert__le,axiom,
    ! [A4: set_real,X: real] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(A4)),finite_card_real(insert_real(X,A4)))) ).

% card_insert_le
tff(fact_2650_card__insert__le,axiom,
    ! [A4: set_nat,X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,insert_nat(X,A4)))) ).

% card_insert_le
tff(fact_2651_card__insert__le,axiom,
    ! [A4: set_complex,X: complex] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A4)),finite_card_complex(insert_complex(X,A4)))) ).

% card_insert_le
tff(fact_2652_card__insert__le,axiom,
    ! [A4: set_int,X: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(A4)),finite_card_int(insert_int(X,A4)))) ).

% card_insert_le
tff(fact_2653_card__insert__le,axiom,
    ! [A4: set_list_nat,X: list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(A4)),finite_card_list_nat(insert_list_nat(X,A4)))) ).

% card_insert_le
tff(fact_2654_card__insert__le,axiom,
    ! [A4: set_set_nat,X: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(A4)),finite_card_set_nat(insert_set_nat(X,A4)))) ).

% card_insert_le
tff(fact_2655_one__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).

% one_le_power
tff(fact_2656_one__le__power,axiom,
    ! [A: real,N: 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(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% one_le_power
tff(fact_2657_one__le__power,axiom,
    ! [A: rat,N: 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),N))) ) ).

% one_le_power
tff(fact_2658_one__le__power,axiom,
    ! [A: nat,N: 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),N))) ) ).

% one_le_power
tff(fact_2659_one__le__power,axiom,
    ! [A: int,N: 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),N))) ) ).

% one_le_power
tff(fact_2660_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y) = one_one_Code_integer )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),N)) = one_one_Code_integer ) ) ).

% left_right_inverse_power
tff(fact_2661_left__right__inverse__power,axiom,
    ! [X: complex,Y: complex,N: 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),N)),aa_nat_complex(power_power_complex(Y),N)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_2662_left__right__inverse__power,axiom,
    ! [X: real,Y: real,N: 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(aa_real_fun_nat_real(power_power_real,X),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_2663_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y) = one_one_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(Y),N)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_2664_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: 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),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_2665_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: 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),N)),aa_nat_int(power_power_int(Y),N)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_2666_finite__enum__subset,axiom,
    ! [X6: set_nat,Y6: set_nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_set_nat_nat(finite_card_nat,X6)))
         => ( aa_nat_nat(infini8530281810654367211te_nat(X6),I3) = aa_nat_nat(infini8530281810654367211te_nat(Y6),I3) ) )
     => ( pp(aa_set_nat_bool(finite_finite_nat,X6))
       => ( pp(aa_set_nat_bool(finite_finite_nat,Y6))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,X6)),aa_set_nat_nat(finite_card_nat,Y6)))
           => pp(aa_set_nat_bool(ord_less_eq_set_nat(X6),Y6)) ) ) ) ) ).

% finite_enum_subset
tff(fact_2667_power__Suc,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),N)) ).

% power_Suc
tff(fact_2668_power__Suc,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% power_Suc
tff(fact_2669_power__Suc,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N)) ).

% power_Suc
tff(fact_2670_power__Suc,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_Suc
tff(fact_2671_power__Suc,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)) ).

% power_Suc
tff(fact_2672_power__Suc2,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),N)),A) ).

% power_Suc2
tff(fact_2673_power__Suc2,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),A) ).

% power_Suc2
tff(fact_2674_power__Suc2,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),N)),A) ).

% power_Suc2
tff(fact_2675_power__Suc2,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) ).

% power_Suc2
tff(fact_2676_power__Suc2,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),A) ).

% power_Suc2
tff(fact_2677_power__0,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),zero_zero_nat) = one_one_Code_integer ).

% power_0
tff(fact_2678_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_2679_power__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_2680_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2681_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_2682_power__add,axiom,
    ! [A: complex,M2: nat,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),M2)),aa_nat_complex(power_power_complex(A),N)) ).

% power_add
tff(fact_2683_power__add,axiom,
    ! [A: real,M2: nat,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% power_add
tff(fact_2684_power__add,axiom,
    ! [A: rat,M2: nat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),M2)),aa_nat_rat(power_power_rat(A),N)) ).

% power_add
tff(fact_2685_power__add,axiom,
    ! [A: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_add
tff(fact_2686_power__add,axiom,
    ! [A: int,M2: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N)) ).

% power_add
tff(fact_2687_nat__power__less__imp__less,axiom,
    ! [I2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),I2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_2688_le__enumerate,axiom,
    ! [S3: set_nat,N: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(infini8530281810654367211te_nat(S3),N))) ) ).

% le_enumerate
tff(fact_2689_card__eq__0__iff,axiom,
    ! [A4: set_list_nat] :
      ( ( finite_card_list_nat(A4) = zero_zero_nat )
    <=> ( ( A4 = bot_bot_set_list_nat )
        | ~ finite8100373058378681591st_nat(A4) ) ) ).

% card_eq_0_iff
tff(fact_2690_card__eq__0__iff,axiom,
    ! [A4: set_set_nat] :
      ( ( finite_card_set_nat(A4) = zero_zero_nat )
    <=> ( ( A4 = bot_bot_set_set_nat )
        | ~ finite1152437895449049373et_nat(A4) ) ) ).

% card_eq_0_iff
tff(fact_2691_card__eq__0__iff,axiom,
    ! [A4: set_complex] :
      ( ( finite_card_complex(A4) = zero_zero_nat )
    <=> ( ( A4 = bot_bot_set_complex )
        | ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4)) ) ) ).

% card_eq_0_iff
tff(fact_2692_card__eq__0__iff,axiom,
    ! [A4: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A4) = zero_zero_nat )
    <=> ( ( A4 = bot_bot_set_nat )
        | ~ pp(aa_set_nat_bool(finite_finite_nat,A4)) ) ) ).

% card_eq_0_iff
tff(fact_2693_card__eq__0__iff,axiom,
    ! [A4: set_int] :
      ( ( finite_card_int(A4) = zero_zero_nat )
    <=> ( ( A4 = bot_bot_set_int )
        | ~ pp(aa_set_int_bool(finite_finite_int,A4)) ) ) ).

% card_eq_0_iff
tff(fact_2694_card__eq__0__iff,axiom,
    ! [A4: set_real] :
      ( ( finite_card_real(A4) = zero_zero_nat )
    <=> ( ( A4 = bot_bot_set_real )
        | ~ pp(aa_set_real_bool(finite_finite_real,A4)) ) ) ).

% card_eq_0_iff
tff(fact_2695_card__ge__0__finite,axiom,
    ! [A4: set_list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_list_nat(A4)))
     => finite8100373058378681591st_nat(A4) ) ).

% card_ge_0_finite
tff(fact_2696_card__ge__0__finite,axiom,
    ! [A4: set_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_set_nat(A4)))
     => finite1152437895449049373et_nat(A4) ) ).

% card_ge_0_finite
tff(fact_2697_card__ge__0__finite,axiom,
    ! [A4: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A4)))
     => pp(aa_set_nat_bool(finite_finite_nat,A4)) ) ).

% card_ge_0_finite
tff(fact_2698_card__ge__0__finite,axiom,
    ! [A4: set_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_int(A4)))
     => pp(aa_set_int_bool(finite_finite_int,A4)) ) ).

% card_ge_0_finite
tff(fact_2699_card__ge__0__finite,axiom,
    ! [A4: set_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_complex(A4)))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,A4)) ) ).

% card_ge_0_finite
tff(fact_2700_card__insert__if,axiom,
    ! [A4: set_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( pp(aa_set_real_bool(member_real(X),A4))
         => ( finite_card_real(insert_real(X,A4)) = finite_card_real(A4) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A4))
         => ( finite_card_real(insert_real(X,A4)) = aa_nat_nat(suc,finite_card_real(A4)) ) ) ) ) ).

% card_insert_if
tff(fact_2701_card__insert__if,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( ( member_list_nat(X,A4)
         => ( finite_card_list_nat(insert_list_nat(X,A4)) = finite_card_list_nat(A4) ) )
        & ( ~ member_list_nat(X,A4)
         => ( finite_card_list_nat(insert_list_nat(X,A4)) = aa_nat_nat(suc,finite_card_list_nat(A4)) ) ) ) ) ).

% card_insert_if
tff(fact_2702_card__insert__if,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( ( member_set_nat(X,A4)
         => ( finite_card_set_nat(insert_set_nat(X,A4)) = finite_card_set_nat(A4) ) )
        & ( ~ member_set_nat(X,A4)
         => ( finite_card_set_nat(insert_set_nat(X,A4)) = aa_nat_nat(suc,finite_card_set_nat(A4)) ) ) ) ) ).

% card_insert_if
tff(fact_2703_card__insert__if,axiom,
    ! [A4: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( pp(aa_set_nat_bool(member_nat(X),A4))
         => ( aa_set_nat_nat(finite_card_nat,insert_nat(X,A4)) = aa_set_nat_nat(finite_card_nat,A4) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
         => ( aa_set_nat_nat(finite_card_nat,insert_nat(X,A4)) = aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,A4)) ) ) ) ) ).

% card_insert_if
tff(fact_2704_card__insert__if,axiom,
    ! [A4: set_int,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( pp(aa_set_int_bool(member_int(X),A4))
         => ( finite_card_int(insert_int(X,A4)) = finite_card_int(A4) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A4))
         => ( finite_card_int(insert_int(X,A4)) = aa_nat_nat(suc,finite_card_int(A4)) ) ) ) ) ).

% card_insert_if
tff(fact_2705_card__insert__if,axiom,
    ! [A4: set_complex,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A4))
         => ( finite_card_complex(insert_complex(X,A4)) = finite_card_complex(A4) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A4))
         => ( finite_card_complex(insert_complex(X,A4)) = aa_nat_nat(suc,finite_card_complex(A4)) ) ) ) ) ).

% card_insert_if
tff(fact_2706_card__Suc__eq__finite,axiom,
    ! [A4: set_real,K2: nat] :
      ( ( finite_card_real(A4) = aa_nat_nat(suc,K2) )
    <=> ? [B4: real,B7: set_real] :
          ( ( A4 = insert_real(B4,B7) )
          & ~ pp(aa_set_real_bool(member_real(B4),B7))
          & ( finite_card_real(B7) = K2 )
          & pp(aa_set_real_bool(finite_finite_real,B7)) ) ) ).

% card_Suc_eq_finite
tff(fact_2707_card__Suc__eq__finite,axiom,
    ! [A4: set_list_nat,K2: nat] :
      ( ( finite_card_list_nat(A4) = aa_nat_nat(suc,K2) )
    <=> ? [B4: list_nat,B7: set_list_nat] :
          ( ( A4 = insert_list_nat(B4,B7) )
          & ~ member_list_nat(B4,B7)
          & ( finite_card_list_nat(B7) = K2 )
          & finite8100373058378681591st_nat(B7) ) ) ).

% card_Suc_eq_finite
tff(fact_2708_card__Suc__eq__finite,axiom,
    ! [A4: set_set_nat,K2: nat] :
      ( ( finite_card_set_nat(A4) = aa_nat_nat(suc,K2) )
    <=> ? [B4: set_nat,B7: set_set_nat] :
          ( ( A4 = insert_set_nat(B4,B7) )
          & ~ member_set_nat(B4,B7)
          & ( finite_card_set_nat(B7) = K2 )
          & finite1152437895449049373et_nat(B7) ) ) ).

% card_Suc_eq_finite
tff(fact_2709_card__Suc__eq__finite,axiom,
    ! [A4: set_nat,K2: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A4) = aa_nat_nat(suc,K2) )
    <=> ? [B4: nat,B7: set_nat] :
          ( ( A4 = insert_nat(B4,B7) )
          & ~ pp(aa_set_nat_bool(member_nat(B4),B7))
          & ( aa_set_nat_nat(finite_card_nat,B7) = K2 )
          & pp(aa_set_nat_bool(finite_finite_nat,B7)) ) ) ).

% card_Suc_eq_finite
tff(fact_2710_card__Suc__eq__finite,axiom,
    ! [A4: set_int,K2: nat] :
      ( ( finite_card_int(A4) = aa_nat_nat(suc,K2) )
    <=> ? [B4: int,B7: set_int] :
          ( ( A4 = insert_int(B4,B7) )
          & ~ pp(aa_set_int_bool(member_int(B4),B7))
          & ( finite_card_int(B7) = K2 )
          & pp(aa_set_int_bool(finite_finite_int,B7)) ) ) ).

% card_Suc_eq_finite
tff(fact_2711_card__Suc__eq__finite,axiom,
    ! [A4: set_complex,K2: nat] :
      ( ( finite_card_complex(A4) = aa_nat_nat(suc,K2) )
    <=> ? [B4: complex,B7: set_complex] :
          ( ( A4 = insert_complex(B4,B7) )
          & ~ pp(aa_set_complex_bool(member_complex(B4),B7))
          & ( finite_card_complex(B7) = K2 )
          & pp(aa_set_complex_bool(finite3207457112153483333omplex,B7)) ) ) ).

% card_Suc_eq_finite
tff(fact_2712_power__less__imp__less__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
     => ( 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_2713_power__less__imp__less__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)))
     => ( 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_2714_power__less__imp__less__base,axiom,
    ! [A: nat,N: 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),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
     => ( 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_2715_power__less__imp__less__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
     => ( 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_2716_card__mono,axiom,
    ! [B5: set_list_nat,A4: set_list_nat] :
      ( finite8100373058378681591st_nat(B5)
     => ( ord_le6045566169113846134st_nat(A4,B5)
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(A4)),finite_card_list_nat(B5))) ) ) ).

% card_mono
tff(fact_2717_card__mono,axiom,
    ! [B5: set_set_nat,A4: set_set_nat] :
      ( finite1152437895449049373et_nat(B5)
     => ( ord_le6893508408891458716et_nat(A4,B5)
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(A4)),finite_card_set_nat(B5))) ) ) ).

% card_mono
tff(fact_2718_card__mono,axiom,
    ! [B5: set_int,A4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B5))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(A4)),finite_card_int(B5))) ) ) ).

% card_mono
tff(fact_2719_card__mono,axiom,
    ! [B5: set_complex,A4: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B5))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A4)),finite_card_complex(B5))) ) ) ).

% card_mono
tff(fact_2720_card__mono,axiom,
    ! [B5: set_nat,A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B5))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A4),B5))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B5))) ) ) ).

% card_mono
tff(fact_2721_card__seteq,axiom,
    ! [B5: set_list_nat,A4: set_list_nat] :
      ( finite8100373058378681591st_nat(B5)
     => ( ord_le6045566169113846134st_nat(A4,B5)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(B5)),finite_card_list_nat(A4)))
         => ( A4 = B5 ) ) ) ) ).

% card_seteq
tff(fact_2722_card__seteq,axiom,
    ! [B5: set_set_nat,A4: set_set_nat] :
      ( finite1152437895449049373et_nat(B5)
     => ( ord_le6893508408891458716et_nat(A4,B5)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(B5)),finite_card_set_nat(A4)))
         => ( A4 = B5 ) ) ) ) ).

% card_seteq
tff(fact_2723_card__seteq,axiom,
    ! [B5: set_int,A4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B5))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(B5)),finite_card_int(A4)))
         => ( A4 = B5 ) ) ) ) ).

% card_seteq
tff(fact_2724_card__seteq,axiom,
    ! [B5: set_complex,A4: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B5))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(B5)),finite_card_complex(A4)))
         => ( A4 = B5 ) ) ) ) ).

% card_seteq
tff(fact_2725_card__seteq,axiom,
    ! [B5: set_nat,A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B5))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A4),B5))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,B5)),aa_set_nat_nat(finite_card_nat,A4)))
         => ( A4 = B5 ) ) ) ) ).

% card_seteq
tff(fact_2726_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_list_nat,C4: nat] :
      ( ! [G: set_list_nat] :
          ( ord_le6045566169113846134st_nat(G,F3)
         => ( finite8100373058378681591st_nat(G)
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(G)),C4)) ) )
     => ( finite8100373058378681591st_nat(F3)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(F3)),C4)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_2727_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_set_nat,C4: nat] :
      ( ! [G: set_set_nat] :
          ( ord_le6893508408891458716et_nat(G,F3)
         => ( finite1152437895449049373et_nat(G)
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(G)),C4)) ) )
     => ( finite1152437895449049373et_nat(F3)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(F3)),C4)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_2728_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_int,C4: nat] :
      ( ! [G: set_int] :
          ( pp(aa_set_int_bool(ord_less_eq_set_int(G),F3))
         => ( pp(aa_set_int_bool(finite_finite_int,G))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(G)),C4)) ) )
     => ( pp(aa_set_int_bool(finite_finite_int,F3))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(F3)),C4)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_2729_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_complex,C4: nat] :
      ( ! [G: set_complex] :
          ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(G),F3))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,G))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(G)),C4)) ) )
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(F3)),C4)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_2730_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_nat,C4: nat] :
      ( ! [G: set_nat] :
          ( pp(aa_set_nat_bool(ord_less_eq_set_nat(G),F3))
         => ( pp(aa_set_nat_bool(finite_finite_nat,G))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,G)),C4)) ) )
     => ( pp(aa_set_nat_bool(finite_finite_nat,F3))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,F3)),C4)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_2731_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_list_nat(S3)))
     => ~ ! [T2: set_list_nat] :
            ( ord_le6045566169113846134st_nat(T2,S3)
           => ( ( finite_card_list_nat(T2) = N )
             => ~ finite8100373058378681591st_nat(T2) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_2732_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_set_nat(S3)))
     => ~ ! [T2: set_set_nat] :
            ( ord_le6893508408891458716et_nat(T2,S3)
           => ( ( finite_card_set_nat(T2) = N )
             => ~ finite1152437895449049373et_nat(T2) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_2733_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_int(S3)))
     => ~ ! [T2: set_int] :
            ( pp(aa_set_int_bool(ord_less_eq_set_int(T2),S3))
           => ( ( finite_card_int(T2) = N )
             => ~ pp(aa_set_int_bool(finite_finite_int,T2)) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_2734_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_complex(S3)))
     => ~ ! [T2: set_complex] :
            ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(T2),S3))
           => ( ( finite_card_complex(T2) = N )
             => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,T2)) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_2735_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_set_nat_nat(finite_card_nat,S3)))
     => ~ ! [T2: set_nat] :
            ( pp(aa_set_nat_bool(ord_less_eq_set_nat(T2),S3))
           => ( ( aa_set_nat_nat(finite_card_nat,T2) = N )
             => ~ pp(aa_set_nat_bool(finite_finite_nat,T2)) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_2736_power__le__one,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),one_one_Code_integer)) ) ) ).

% power_le_one
tff(fact_2737_power__le__one,axiom,
    ! [A: real,N: 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(aa_real_fun_nat_real(power_power_real,A),N)),one_one_real)) ) ) ).

% power_le_one
tff(fact_2738_power__le__one,axiom,
    ! [A: rat,N: 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),N)),one_one_rat)) ) ) ).

% power_le_one
tff(fact_2739_power__le__one,axiom,
    ! [A: nat,N: 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),N)),one_one_nat)) ) ) ).

% power_le_one
tff(fact_2740_power__le__one,axiom,
    ! [A: int,N: 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),N)),one_one_int)) ) ) ).

% power_le_one
tff(fact_2741_power__inject__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,N)) )
     => ( 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_2742_power__inject__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N)) )
     => ( 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_2743_power__inject__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N)) )
     => ( 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_2744_power__inject__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,N)) )
     => ( 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_2745_power__le__imp__le__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,N))))
     => ( 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_2746_power__le__imp__le__base,axiom,
    ! [A: rat,N: 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,N))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N))))
     => ( 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_2747_power__le__imp__le__base,axiom,
    ! [A: nat,N: 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,N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N))))
     => ( 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_2748_power__le__imp__le__base,axiom,
    ! [A: int,N: 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,N))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,N))))
     => ( 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_2749_power__less__power__Suc,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))) ) ).

% power_less_power_Suc
tff(fact_2750_power__less__power__Suc,axiom,
    ! [A: real,N: 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(aa_real_fun_nat_real(power_power_real,A),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))) ) ).

% power_less_power_Suc
tff(fact_2751_power__less__power__Suc,axiom,
    ! [A: rat,N: 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),N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N)))) ) ).

% power_less_power_Suc
tff(fact_2752_power__less__power__Suc,axiom,
    ! [A: nat,N: 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),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))) ) ).

% power_less_power_Suc
tff(fact_2753_power__less__power__Suc,axiom,
    ! [A: int,N: 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),N)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)))) ) ).

% power_less_power_Suc
tff(fact_2754_power__gt1__lemma,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))) ) ).

% power_gt1_lemma
tff(fact_2755_power__gt1__lemma,axiom,
    ! [A: real,N: 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(aa_real_fun_nat_real(power_power_real,A),N)))) ) ).

% power_gt1_lemma
tff(fact_2756_power__gt1__lemma,axiom,
    ! [A: rat,N: 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(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N)))) ) ).

% power_gt1_lemma
tff(fact_2757_power__gt1__lemma,axiom,
    ! [A: nat,N: 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),N)))) ) ).

% power_gt1_lemma
tff(fact_2758_power__gt1__lemma,axiom,
    ! [A: int,N: 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),N)))) ) ).

% power_gt1_lemma
tff(fact_2759_card__le__sym__Diff,axiom,
    ! [A4: set_list_nat,B5: set_list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( finite8100373058378681591st_nat(B5)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(A4)),finite_card_list_nat(B5)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A4,B5))),finite_card_list_nat(minus_7954133019191499631st_nat(B5,A4)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_2760_card__le__sym__Diff,axiom,
    ! [A4: set_set_nat,B5: set_set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( finite1152437895449049373et_nat(B5)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(A4)),finite_card_set_nat(B5)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A4,B5))),finite_card_set_nat(minus_2163939370556025621et_nat(B5,A4)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_2761_card__le__sym__Diff,axiom,
    ! [A4: set_int,B5: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(A4)),finite_card_int(B5)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(minus_minus_set_int(A4,B5))),finite_card_int(minus_minus_set_int(B5,A4)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_2762_card__le__sym__Diff,axiom,
    ! [A4: set_complex,B5: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A4)),finite_card_complex(B5)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(minus_811609699411566653omplex(A4,B5))),finite_card_complex(minus_811609699411566653omplex(B5,A4)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_2763_card__le__sym__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( finite6177210948735845034at_nat(A4)
     => ( finite6177210948735845034at_nat(B5)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A4)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B5)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B5))),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B5),A4)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_2764_card__le__sym__Diff,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B5)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B5))),aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B5),A4)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_2765_card__length,axiom,
    ! [Xs: list_complex] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(set_complex2(Xs))),size_s3451745648224563538omplex(Xs))) ).

% card_length
tff(fact_2766_card__length,axiom,
    ! [Xs: list_list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(set_list_nat2(Xs))),size_s3023201423986296836st_nat(Xs))) ).

% card_length
tff(fact_2767_card__length,axiom,
    ! [Xs: list_set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(set_set_nat2(Xs))),size_s3254054031482475050et_nat(Xs))) ).

% card_length
tff(fact_2768_card__length,axiom,
    ! [Xs: list_VEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite7802652506058667612T_VEBT(set_VEBT_VEBT2(Xs))),size_s6755466524823107622T_VEBT(Xs))) ).

% card_length
tff(fact_2769_card__length,axiom,
    ! [Xs: list_bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_bool(set_bool2(Xs))),size_size_list_bool(Xs))) ).

% card_length
tff(fact_2770_card__length,axiom,
    ! [Xs: list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,set_nat2(Xs))),aa_list_nat_nat(size_size_list_nat,Xs))) ).

% card_length
tff(fact_2771_card__length,axiom,
    ! [Xs: list_int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(set_int2(Xs))),size_size_list_int(Xs))) ).

% card_length
tff(fact_2772_power__gt1,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_2773_power__gt1,axiom,
    ! [A: real,N: 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(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_2774_power__gt1,axiom,
    ! [A: rat,N: 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,N)))) ) ).

% power_gt1
tff(fact_2775_power__gt1,axiom,
    ! [A: nat,N: 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,N)))) ) ).

% power_gt1
tff(fact_2776_power__gt1,axiom,
    ! [A: int,N: 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,N)))) ) ).

% power_gt1
tff(fact_2777_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = zero_z3403309356797280102nteger ) ) ) ).

% power_0_left
tff(fact_2778_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).

% power_0_left
tff(fact_2779_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).

% power_0_left
tff(fact_2780_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = zero_zero_real ) ) ) ).

% power_0_left
tff(fact_2781_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ) ).

% power_0_left
tff(fact_2782_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% power_0_left
tff(fact_2783_power__increasing,axiom,
    ! [N: nat,N5: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N5))) ) ) ).

% power_increasing
tff(fact_2784_power__increasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),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(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N5))) ) ) ).

% power_increasing
tff(fact_2785_power__increasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),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),N)),aa_nat_rat(power_power_rat(A),N5))) ) ) ).

% power_increasing
tff(fact_2786_power__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),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),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5))) ) ) ).

% power_increasing
tff(fact_2787_power__increasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),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),N)),aa_nat_int(power_power_int(A),N5))) ) ) ).

% power_increasing
tff(fact_2788_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ).

% zero_power
tff(fact_2789_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ).

% zero_power
tff(fact_2790_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = zero_zero_real ) ) ).

% zero_power
tff(fact_2791_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ).

% zero_power
tff(fact_2792_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ).

% zero_power
tff(fact_2793_lenlex__length,axiom,
    ! [Ms: list_VEBT_VEBT,Ns: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Ms,Ns),lenlex_VEBT_VEBT(R))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Ms)),size_s6755466524823107622T_VEBT(Ns))) ) ).

% lenlex_length
tff(fact_2794_lenlex__length,axiom,
    ! [Ms: list_bool,Ns: list_bool,R: set_Pr8621012493645324487l_bool] :
      ( member6300827366518646928t_bool(produc970142415094439895t_bool(Ms,Ns),lenlex_bool(R))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Ms)),size_size_list_bool(Ns))) ) ).

% lenlex_length
tff(fact_2795_lenlex__length,axiom,
    ! [Ms: list_nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Ms,Ns),lenlex_nat(R))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Ms)),aa_list_nat_nat(size_size_list_nat,Ns))) ) ).

% lenlex_length
tff(fact_2796_lenlex__length,axiom,
    ! [Ms: list_int,Ns: list_int,R: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Ms,Ns),lenlex_int(R))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Ms)),size_size_list_int(Ns))) ) ).

% lenlex_length
tff(fact_2797_power__gt__expt,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K2))) ) ).

% power_gt_expt
tff(fact_2798_nat__one__le__power,axiom,
    ! [I2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I2))
     => 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,I2),N))) ) ).

% nat_one_le_power
tff(fact_2799_enumerate__step,axiom,
    ! [S3: set_nat,N: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S3),N)),aa_nat_nat(infini8530281810654367211te_nat(S3),aa_nat_nat(suc,N)))) ) ).

% enumerate_step
tff(fact_2800_card__gt__0__iff,axiom,
    ! [A4: set_list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_list_nat(A4)))
    <=> ( ( A4 != bot_bot_set_list_nat )
        & finite8100373058378681591st_nat(A4) ) ) ).

% card_gt_0_iff
tff(fact_2801_card__gt__0__iff,axiom,
    ! [A4: set_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_set_nat(A4)))
    <=> ( ( A4 != bot_bot_set_set_nat )
        & finite1152437895449049373et_nat(A4) ) ) ).

% card_gt_0_iff
tff(fact_2802_card__gt__0__iff,axiom,
    ! [A4: set_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_complex(A4)))
    <=> ( ( A4 != bot_bot_set_complex )
        & pp(aa_set_complex_bool(finite3207457112153483333omplex,A4)) ) ) ).

% card_gt_0_iff
tff(fact_2803_card__gt__0__iff,axiom,
    ! [A4: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A4)))
    <=> ( ( A4 != bot_bot_set_nat )
        & pp(aa_set_nat_bool(finite_finite_nat,A4)) ) ) ).

% card_gt_0_iff
tff(fact_2804_card__gt__0__iff,axiom,
    ! [A4: set_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_int(A4)))
    <=> ( ( A4 != bot_bot_set_int )
        & pp(aa_set_int_bool(finite_finite_int,A4)) ) ) ).

% card_gt_0_iff
tff(fact_2805_card__gt__0__iff,axiom,
    ! [A4: set_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_real(A4)))
    <=> ( ( A4 != bot_bot_set_real )
        & pp(aa_set_real_bool(finite_finite_real,A4)) ) ) ).

% card_gt_0_iff
tff(fact_2806_card__1__singleton__iff,axiom,
    ! [A4: set_complex] :
      ( ( finite_card_complex(A4) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X4: complex] : A4 = insert_complex(X4,bot_bot_set_complex) ) ).

% card_1_singleton_iff
tff(fact_2807_card__1__singleton__iff,axiom,
    ! [A4: set_list_nat] :
      ( ( finite_card_list_nat(A4) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X4: list_nat] : A4 = insert_list_nat(X4,bot_bot_set_list_nat) ) ).

% card_1_singleton_iff
tff(fact_2808_card__1__singleton__iff,axiom,
    ! [A4: set_set_nat] :
      ( ( finite_card_set_nat(A4) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X4: set_nat] : A4 = insert_set_nat(X4,bot_bot_set_set_nat) ) ).

% card_1_singleton_iff
tff(fact_2809_card__1__singleton__iff,axiom,
    ! [A4: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A4) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X4: nat] : A4 = insert_nat(X4,bot_bot_set_nat) ) ).

% card_1_singleton_iff
tff(fact_2810_card__1__singleton__iff,axiom,
    ! [A4: set_int] :
      ( ( finite_card_int(A4) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X4: int] : A4 = insert_int(X4,bot_bot_set_int) ) ).

% card_1_singleton_iff
tff(fact_2811_card__1__singleton__iff,axiom,
    ! [A4: set_real] :
      ( ( finite_card_real(A4) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X4: real] : A4 = insert_real(X4,bot_bot_set_real) ) ).

% card_1_singleton_iff
tff(fact_2812_card__eq__SucD,axiom,
    ! [A4: set_complex,K2: nat] :
      ( ( finite_card_complex(A4) = aa_nat_nat(suc,K2) )
     => ? [B3: complex,B8: set_complex] :
          ( ( A4 = insert_complex(B3,B8) )
          & ~ pp(aa_set_complex_bool(member_complex(B3),B8))
          & ( finite_card_complex(B8) = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B8 = bot_bot_set_complex ) ) ) ) ).

% card_eq_SucD
tff(fact_2813_card__eq__SucD,axiom,
    ! [A4: set_list_nat,K2: nat] :
      ( ( finite_card_list_nat(A4) = aa_nat_nat(suc,K2) )
     => ? [B3: list_nat,B8: set_list_nat] :
          ( ( A4 = insert_list_nat(B3,B8) )
          & ~ member_list_nat(B3,B8)
          & ( finite_card_list_nat(B8) = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B8 = bot_bot_set_list_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_2814_card__eq__SucD,axiom,
    ! [A4: set_set_nat,K2: nat] :
      ( ( finite_card_set_nat(A4) = aa_nat_nat(suc,K2) )
     => ? [B3: set_nat,B8: set_set_nat] :
          ( ( A4 = insert_set_nat(B3,B8) )
          & ~ member_set_nat(B3,B8)
          & ( finite_card_set_nat(B8) = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B8 = bot_bot_set_set_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_2815_card__eq__SucD,axiom,
    ! [A4: set_nat,K2: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A4) = aa_nat_nat(suc,K2) )
     => ? [B3: nat,B8: set_nat] :
          ( ( A4 = insert_nat(B3,B8) )
          & ~ pp(aa_set_nat_bool(member_nat(B3),B8))
          & ( aa_set_nat_nat(finite_card_nat,B8) = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B8 = bot_bot_set_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_2816_card__eq__SucD,axiom,
    ! [A4: set_int,K2: nat] :
      ( ( finite_card_int(A4) = aa_nat_nat(suc,K2) )
     => ? [B3: int,B8: set_int] :
          ( ( A4 = insert_int(B3,B8) )
          & ~ pp(aa_set_int_bool(member_int(B3),B8))
          & ( finite_card_int(B8) = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B8 = bot_bot_set_int ) ) ) ) ).

% card_eq_SucD
tff(fact_2817_card__eq__SucD,axiom,
    ! [A4: set_real,K2: nat] :
      ( ( finite_card_real(A4) = aa_nat_nat(suc,K2) )
     => ? [B3: real,B8: set_real] :
          ( ( A4 = insert_real(B3,B8) )
          & ~ pp(aa_set_real_bool(member_real(B3),B8))
          & ( finite_card_real(B8) = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B8 = bot_bot_set_real ) ) ) ) ).

% card_eq_SucD
tff(fact_2818_card__Suc__eq,axiom,
    ! [A4: set_complex,K2: nat] :
      ( ( finite_card_complex(A4) = aa_nat_nat(suc,K2) )
    <=> ? [B4: complex,B7: set_complex] :
          ( ( A4 = insert_complex(B4,B7) )
          & ~ pp(aa_set_complex_bool(member_complex(B4),B7))
          & ( finite_card_complex(B7) = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B7 = bot_bot_set_complex ) ) ) ) ).

% card_Suc_eq
tff(fact_2819_card__Suc__eq,axiom,
    ! [A4: set_list_nat,K2: nat] :
      ( ( finite_card_list_nat(A4) = aa_nat_nat(suc,K2) )
    <=> ? [B4: list_nat,B7: set_list_nat] :
          ( ( A4 = insert_list_nat(B4,B7) )
          & ~ member_list_nat(B4,B7)
          & ( finite_card_list_nat(B7) = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B7 = bot_bot_set_list_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_2820_card__Suc__eq,axiom,
    ! [A4: set_set_nat,K2: nat] :
      ( ( finite_card_set_nat(A4) = aa_nat_nat(suc,K2) )
    <=> ? [B4: set_nat,B7: set_set_nat] :
          ( ( A4 = insert_set_nat(B4,B7) )
          & ~ member_set_nat(B4,B7)
          & ( finite_card_set_nat(B7) = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B7 = bot_bot_set_set_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_2821_card__Suc__eq,axiom,
    ! [A4: set_nat,K2: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A4) = aa_nat_nat(suc,K2) )
    <=> ? [B4: nat,B7: set_nat] :
          ( ( A4 = insert_nat(B4,B7) )
          & ~ pp(aa_set_nat_bool(member_nat(B4),B7))
          & ( aa_set_nat_nat(finite_card_nat,B7) = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B7 = bot_bot_set_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_2822_card__Suc__eq,axiom,
    ! [A4: set_int,K2: nat] :
      ( ( finite_card_int(A4) = aa_nat_nat(suc,K2) )
    <=> ? [B4: int,B7: set_int] :
          ( ( A4 = insert_int(B4,B7) )
          & ~ pp(aa_set_int_bool(member_int(B4),B7))
          & ( finite_card_int(B7) = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B7 = bot_bot_set_int ) ) ) ) ).

% card_Suc_eq
tff(fact_2823_card__Suc__eq,axiom,
    ! [A4: set_real,K2: nat] :
      ( ( finite_card_real(A4) = aa_nat_nat(suc,K2) )
    <=> ? [B4: real,B7: set_real] :
          ( ( A4 = insert_real(B4,B7) )
          & ~ pp(aa_set_real_bool(member_real(B4),B7))
          & ( finite_card_real(B7) = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B7 = bot_bot_set_real ) ) ) ) ).

% card_Suc_eq
tff(fact_2824_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(A4)),aa_nat_nat(suc,zero_zero_nat)))
      <=> ! [X4: list_nat] :
            ( member_list_nat(X4,A4)
           => ! [Xa3: list_nat] :
                ( member_list_nat(Xa3,A4)
               => ( X4 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_2825_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(A4)),aa_nat_nat(suc,zero_zero_nat)))
      <=> ! [X4: set_nat] :
            ( member_set_nat(X4,A4)
           => ! [Xa3: set_nat] :
                ( member_set_nat(Xa3,A4)
               => ( X4 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_2826_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_nat_nat(suc,zero_zero_nat)))
      <=> ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A4))
           => ! [Xa3: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa3),A4))
               => ( X4 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_2827_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(A4)),aa_nat_nat(suc,zero_zero_nat)))
      <=> ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
           => ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),A4))
               => ( X4 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_2828_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A4)),aa_nat_nat(suc,zero_zero_nat)))
      <=> ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A4))
           => ! [Xa3: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa3),A4))
               => ( X4 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_2829_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),finite_card_real(A4)))
    <=> ? [A5: real,B7: set_real] :
          ( ( A4 = insert_real(A5,B7) )
          & ~ pp(aa_set_real_bool(member_real(A5),B7))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_real(B7)))
          & pp(aa_set_real_bool(finite_finite_real,B7)) ) ) ).

% card_le_Suc_iff
tff(fact_2830_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),finite_card_list_nat(A4)))
    <=> ? [A5: list_nat,B7: set_list_nat] :
          ( ( A4 = insert_list_nat(A5,B7) )
          & ~ member_list_nat(A5,B7)
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_list_nat(B7)))
          & finite8100373058378681591st_nat(B7) ) ) ).

% card_le_Suc_iff
tff(fact_2831_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),finite_card_set_nat(A4)))
    <=> ? [A5: set_nat,B7: set_set_nat] :
          ( ( A4 = insert_set_nat(A5,B7) )
          & ~ member_set_nat(A5,B7)
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_set_nat(B7)))
          & finite1152437895449049373et_nat(B7) ) ) ).

% card_le_Suc_iff
tff(fact_2832_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_set_nat_nat(finite_card_nat,A4)))
    <=> ? [A5: nat,B7: set_nat] :
          ( ( A4 = insert_nat(A5,B7) )
          & ~ pp(aa_set_nat_bool(member_nat(A5),B7))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_set_nat_nat(finite_card_nat,B7)))
          & pp(aa_set_nat_bool(finite_finite_nat,B7)) ) ) ).

% card_le_Suc_iff
tff(fact_2833_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),finite_card_int(A4)))
    <=> ? [A5: int,B7: set_int] :
          ( ( A4 = insert_int(A5,B7) )
          & ~ pp(aa_set_int_bool(member_int(A5),B7))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_int(B7)))
          & pp(aa_set_int_bool(finite_finite_int,B7)) ) ) ).

% card_le_Suc_iff
tff(fact_2834_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),finite_card_complex(A4)))
    <=> ? [A5: complex,B7: set_complex] :
          ( ( A4 = insert_complex(A5,B7) )
          & ~ pp(aa_set_complex_bool(member_complex(A5),B7))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),finite_card_complex(B7)))
          & pp(aa_set_complex_bool(finite3207457112153483333omplex,B7)) ) ) ).

% card_le_Suc_iff
tff(fact_2835_power__Suc__less,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ) ).

% power_Suc_less
tff(fact_2836_power__Suc__less,axiom,
    ! [A: real,N: 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(aa_real_fun_nat_real(power_power_real,A),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ).

% power_Suc_less
tff(fact_2837_power__Suc__less,axiom,
    ! [A: rat,N: 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(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N))),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% power_Suc_less
tff(fact_2838_power__Suc__less,axiom,
    ! [A: nat,N: 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),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% power_Suc_less
tff(fact_2839_power__Suc__less,axiom,
    ! [A: int,N: 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),N))),aa_nat_int(power_power_int(A),N))) ) ) ).

% power_Suc_less
tff(fact_2840_power__Suc__le__self,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2841_power__Suc__le__self,axiom,
    ! [A: real,N: 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(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2842_power__Suc__le__self,axiom,
    ! [A: rat,N: 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,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2843_power__Suc__le__self,axiom,
    ! [A: nat,N: 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,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2844_power__Suc__le__self,axiom,
    ! [A: int,N: 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,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2845_power__Suc__less__one,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))),one_one_Code_integer)) ) ) ).

% power_Suc_less_one
tff(fact_2846_power__Suc__less__one,axiom,
    ! [A: real,N: 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(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N))),one_one_real)) ) ) ).

% power_Suc_less_one
tff(fact_2847_power__Suc__less__one,axiom,
    ! [A: rat,N: 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,N))),one_one_rat)) ) ) ).

% power_Suc_less_one
tff(fact_2848_power__Suc__less__one,axiom,
    ! [A: nat,N: 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,N))),one_one_nat)) ) ) ).

% power_Suc_less_one
tff(fact_2849_power__Suc__less__one,axiom,
    ! [A: int,N: 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,N))),one_one_int)) ) ) ).

% power_Suc_less_one
tff(fact_2850_card__Diff1__le,axiom,
    ! [A4: set_complex,X: complex] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))),finite_card_complex(A4))) ).

% card_Diff1_le
tff(fact_2851_card__Diff1__le,axiom,
    ! [A4: set_list_nat,X: list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A4,insert_list_nat(X,bot_bot_set_list_nat)))),finite_card_list_nat(A4))) ).

% card_Diff1_le
tff(fact_2852_card__Diff1__le,axiom,
    ! [A4: set_set_nat,X: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A4,insert_set_nat(X,bot_bot_set_set_nat)))),finite_card_set_nat(A4))) ).

% card_Diff1_le
tff(fact_2853_card__Diff1__le,axiom,
    ! [A4: set_int,X: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))),finite_card_int(A4))) ).

% card_Diff1_le
tff(fact_2854_card__Diff1__le,axiom,
    ! [A4: set_real,X: real] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))),finite_card_real(A4))) ).

% card_Diff1_le
tff(fact_2855_card__Diff1__le,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)))),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A4))) ).

% card_Diff1_le
tff(fact_2856_card__Diff1__le,axiom,
    ! [A4: set_nat,X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X,bot_bot_set_nat)))),aa_set_nat_nat(finite_card_nat,A4))) ).

% card_Diff1_le
tff(fact_2857_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N5))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),one_one_Code_integer))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N5)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2858_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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(aa_real_fun_nat_real(power_power_real,A),N5)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2859_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2860_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2861_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_2862_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N5)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ) ) ).

% power_decreasing
tff(fact_2863_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),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(aa_real_fun_nat_real(power_power_real,A),N5)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ) ).

% power_decreasing
tff(fact_2864_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),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),N))) ) ) ) ).

% power_decreasing
tff(fact_2865_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),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),N))) ) ) ) ).

% power_decreasing
tff(fact_2866_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),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),N))) ) ) ) ).

% power_decreasing
tff(fact_2867_power__le__imp__le__exp,axiom,
    ! [A: code_integer,M2: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M2)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2868_power__le__imp__le__exp,axiom,
    ! [A: real,M2: nat,N: 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(aa_real_fun_nat_real(power_power_real,A),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2869_power__le__imp__le__exp,axiom,
    ! [A: rat,M2: nat,N: 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),M2)),aa_nat_rat(power_power_rat(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2870_power__le__imp__le__exp,axiom,
    ! [A: nat,M2: nat,N: 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),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2871_power__le__imp__le__exp,axiom,
    ! [A: int,M2: nat,N: 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),M2)),aa_nat_int(power_power_int(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2872_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: real,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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(aa_real_fun_nat_real(power_power_real,A),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2873_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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),N) = aa_nat_rat(power_power_rat(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2874_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B: 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(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),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2875_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: int,B: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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),N) = aa_nat_int(power_power_int(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2876_power__eq__imp__eq__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N) )
     => ( 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),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2877_power__eq__imp__eq__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
     => ( 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),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2878_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
     => ( 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),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2879_power__eq__imp__eq__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B),N) )
     => ( 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),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2880_diff__card__le__card__Diff,axiom,
    ! [B5: set_list_nat,A4: set_list_nat] :
      ( finite8100373058378681591st_nat(B5)
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A4)),finite_card_list_nat(B5))),finite_card_list_nat(minus_7954133019191499631st_nat(A4,B5)))) ) ).

% diff_card_le_card_Diff
tff(fact_2881_diff__card__le__card__Diff,axiom,
    ! [B5: set_set_nat,A4: set_set_nat] :
      ( finite1152437895449049373et_nat(B5)
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A4)),finite_card_set_nat(B5))),finite_card_set_nat(minus_2163939370556025621et_nat(A4,B5)))) ) ).

% diff_card_le_card_Diff
tff(fact_2882_diff__card__le__card__Diff,axiom,
    ! [B5: set_int,A4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B5))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A4)),finite_card_int(B5))),finite_card_int(minus_minus_set_int(A4,B5)))) ) ).

% diff_card_le_card_Diff
tff(fact_2883_diff__card__le__card__Diff,axiom,
    ! [B5: set_complex,A4: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A4)),finite_card_complex(B5))),finite_card_complex(minus_811609699411566653omplex(A4,B5)))) ) ).

% diff_card_le_card_Diff
tff(fact_2884_diff__card__le__card__Diff,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( finite6177210948735845034at_nat(B5)
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A4)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B5))),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B5)))) ) ).

% diff_card_le_card_Diff
tff(fact_2885_diff__card__le__card__Diff,axiom,
    ! [B5: set_nat,A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B5))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B5))),aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B5)))) ) ).

% diff_card_le_card_Diff
tff(fact_2886_self__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ) ).

% self_le_power
tff(fact_2887_self__le__power,axiom,
    ! [A: real,N: 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),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ).

% self_le_power
tff(fact_2888_self__le__power,axiom,
    ! [A: rat,N: 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),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% self_le_power
tff(fact_2889_self__le__power,axiom,
    ! [A: nat,N: 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),N))
       => 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),N))) ) ) ).

% self_le_power
tff(fact_2890_self__le__power,axiom,
    ! [A: int,N: 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),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(A),N))) ) ) ).

% self_le_power
tff(fact_2891_one__less__power,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ) ).

% one_less_power
tff(fact_2892_one__less__power,axiom,
    ! [A: real,N: 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),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ).

% one_less_power
tff(fact_2893_one__less__power,axiom,
    ! [A: rat,N: 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),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% one_less_power
tff(fact_2894_one__less__power,axiom,
    ! [A: nat,N: 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),N))
       => 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),N))) ) ) ).

% one_less_power
tff(fact_2895_one__less__power,axiom,
    ! [A: int,N: 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),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),N))) ) ) ).

% one_less_power
tff(fact_2896_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_2897_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_2898_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)),aa_list_nat_nat(size_size_list_nat,Xs))) ).

% count_le_length
tff(fact_2899_count__le__length,axiom,
    ! [Xs: list_int,X: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(count_list_int(Xs),X)),size_size_list_int(Xs))) ).

% count_le_length
tff(fact_2900_dbl__inc__def,axiom,
    ! [X: code_integer] : neg_nu5831290666863070958nteger(X) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(X,X),one_one_Code_integer) ).

% dbl_inc_def
tff(fact_2901_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_2902_dbl__inc__def,axiom,
    ! [X: real] : aa_real_real(neg_nu8295874005876285629c_real,X) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(X),X)),one_one_real) ).

% dbl_inc_def
tff(fact_2903_dbl__inc__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,X) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X)),one_one_rat) ).

% dbl_inc_def
tff(fact_2904_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_2905_card_Oremove,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( member_list_nat(X,A4)
       => ( finite_card_list_nat(A4) = aa_nat_nat(suc,finite_card_list_nat(minus_7954133019191499631st_nat(A4,insert_list_nat(X,bot_bot_set_list_nat)))) ) ) ) ).

% card.remove
tff(fact_2906_card_Oremove,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( member_set_nat(X,A4)
       => ( finite_card_set_nat(A4) = aa_nat_nat(suc,finite_card_set_nat(minus_2163939370556025621et_nat(A4,insert_set_nat(X,bot_bot_set_set_nat)))) ) ) ) ).

% card.remove
tff(fact_2907_card_Oremove,axiom,
    ! [A4: set_complex,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(member_complex(X),A4))
       => ( finite_card_complex(A4) = aa_nat_nat(suc,finite_card_complex(minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% card.remove
tff(fact_2908_card_Oremove,axiom,
    ! [A4: set_int,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(X),A4))
       => ( finite_card_int(A4) = aa_nat_nat(suc,finite_card_int(minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% card.remove
tff(fact_2909_card_Oremove,axiom,
    ! [A4: set_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(X),A4))
       => ( finite_card_real(A4) = aa_nat_nat(suc,finite_card_real(minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% card.remove
tff(fact_2910_card_Oremove,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( finite6177210948735845034at_nat(A4)
     => ( member8440522571783428010at_nat(X,A4)
       => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A4) = aa_nat_nat(suc,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)))) ) ) ) ).

% card.remove
tff(fact_2911_card_Oremove,axiom,
    ! [A4: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(member_nat(X),A4))
       => ( aa_set_nat_nat(finite_card_nat,A4) = aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X,bot_bot_set_nat)))) ) ) ) ).

% card.remove
tff(fact_2912_card_Oinsert__remove,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( finite_card_list_nat(insert_list_nat(X,A4)) = aa_nat_nat(suc,finite_card_list_nat(minus_7954133019191499631st_nat(A4,insert_list_nat(X,bot_bot_set_list_nat)))) ) ) ).

% card.insert_remove
tff(fact_2913_card_Oinsert__remove,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( finite_card_set_nat(insert_set_nat(X,A4)) = aa_nat_nat(suc,finite_card_set_nat(minus_2163939370556025621et_nat(A4,insert_set_nat(X,bot_bot_set_set_nat)))) ) ) ).

% card.insert_remove
tff(fact_2914_card_Oinsert__remove,axiom,
    ! [A4: set_complex,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( finite_card_complex(insert_complex(X,A4)) = aa_nat_nat(suc,finite_card_complex(minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% card.insert_remove
tff(fact_2915_card_Oinsert__remove,axiom,
    ! [A4: set_int,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( finite_card_int(insert_int(X,A4)) = aa_nat_nat(suc,finite_card_int(minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) ) ) ).

% card.insert_remove
tff(fact_2916_card_Oinsert__remove,axiom,
    ! [A4: set_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( finite_card_real(insert_real(X,A4)) = aa_nat_nat(suc,finite_card_real(minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ).

% card.insert_remove
tff(fact_2917_card_Oinsert__remove,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( finite6177210948735845034at_nat(A4)
     => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,insert8211810215607154385at_nat(X,A4)) = aa_nat_nat(suc,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)))) ) ) ).

% card.insert_remove
tff(fact_2918_card_Oinsert__remove,axiom,
    ! [A4: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( aa_set_nat_nat(finite_card_nat,insert_nat(X,A4)) = aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X,bot_bot_set_nat)))) ) ) ).

% card.insert_remove
tff(fact_2919_card__Suc__Diff1,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( member_list_nat(X,A4)
       => ( aa_nat_nat(suc,finite_card_list_nat(minus_7954133019191499631st_nat(A4,insert_list_nat(X,bot_bot_set_list_nat)))) = finite_card_list_nat(A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_2920_card__Suc__Diff1,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( member_set_nat(X,A4)
       => ( aa_nat_nat(suc,finite_card_set_nat(minus_2163939370556025621et_nat(A4,insert_set_nat(X,bot_bot_set_set_nat)))) = finite_card_set_nat(A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_2921_card__Suc__Diff1,axiom,
    ! [A4: set_complex,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(member_complex(X),A4))
       => ( aa_nat_nat(suc,finite_card_complex(minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) = finite_card_complex(A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_2922_card__Suc__Diff1,axiom,
    ! [A4: set_int,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(X),A4))
       => ( aa_nat_nat(suc,finite_card_int(minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) = finite_card_int(A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_2923_card__Suc__Diff1,axiom,
    ! [A4: set_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(X),A4))
       => ( aa_nat_nat(suc,finite_card_real(minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) = finite_card_real(A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_2924_card__Suc__Diff1,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( finite6177210948735845034at_nat(A4)
     => ( member8440522571783428010at_nat(X,A4)
       => ( aa_nat_nat(suc,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)))) = aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_2925_card__Suc__Diff1,axiom,
    ! [A4: set_nat,X: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(member_nat(X),A4))
       => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X,bot_bot_set_nat)))) = aa_set_nat_nat(finite_card_nat,A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_2926_power__strict__mono,axiom,
    ! [A: real,B: real,N: 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),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ) ).

% power_strict_mono
tff(fact_2927_power__strict__mono,axiom,
    ! [A: rat,B: rat,N: 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),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ) ).

% power_strict_mono
tff(fact_2928_power__strict__mono,axiom,
    ! [A: nat,B: nat,N: 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),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))) ) ) ) ).

% power_strict_mono
tff(fact_2929_power__strict__mono,axiom,
    ! [A: int,B: int,N: 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),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ) ).

% power_strict_mono
tff(fact_2930_power__eq__if,axiom,
    ! [M2: nat,P: code_integer] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(P),M2) = one_one_Code_integer ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(P),M2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,P),aa_nat_Code_integer(power_8256067586552552935nteger(P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2931_power__eq__if,axiom,
    ! [M2: nat,P: complex] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P),M2) = one_one_complex ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P),M2) = aa_complex_complex(times_times_complex(P),aa_nat_complex(power_power_complex(P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2932_power__eq__if,axiom,
    ! [M2: nat,P: real] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,P),M2) = one_one_real ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,P),M2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P),aa_nat_real(aa_real_fun_nat_real(power_power_real,P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2933_power__eq__if,axiom,
    ! [M2: nat,P: rat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P),M2) = one_one_rat ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P),M2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P),aa_nat_rat(power_power_rat(P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2934_power__eq__if,axiom,
    ! [M2: nat,P: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P),M2) = one_one_nat ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P),M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2935_power__eq__if,axiom,
    ! [M2: nat,P: int] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_int(power_power_int(P),M2) = one_one_int ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_int(power_power_int(P),M2) = aa_int_int(aa_int_fun_int_int(times_times_int,P),aa_nat_int(power_power_int(P),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2936_power__minus__mult,axiom,
    ! [N: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),N) ) ) ).

% power_minus_mult
tff(fact_2937_power__minus__mult,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) ) ) ).

% power_minus_mult
tff(fact_2938_power__minus__mult,axiom,
    ! [N: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),N) ) ) ).

% power_minus_mult
tff(fact_2939_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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,N),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) ) ) ).

% power_minus_mult
tff(fact_2940_power__minus__mult,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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,N),one_one_nat))),A) = aa_nat_int(power_power_int(A),N) ) ) ).

% power_minus_mult
tff(fact_2941_Cons__in__lex,axiom,
    ! [X: code_integer,Xs: list_Code_integer,Y: code_integer,Ys: list_Code_integer,R: set_Pr4811707699266497531nteger] :
      ( member749217712838834276nteger(produc750622340256944499nteger(cons_Code_integer(X,Xs),cons_Code_integer(Y,Ys)),lex_Code_integer(R))
    <=> ( ( member157494554546826820nteger(produc1086072967326762835nteger(X,Y),R)
          & ( size_s3445333598471063425nteger(Xs) = size_s3445333598471063425nteger(Ys) ) )
        | ( ( X = Y )
          & member749217712838834276nteger(produc750622340256944499nteger(Xs,Ys),lex_Code_integer(R)) ) ) ) ).

% Cons_in_lex
tff(fact_2942_Cons__in__lex,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
      ( member6693912407220327184at_nat(produc5943733680697469783at_nat(cons_P6512896166579812791at_nat(X,Xs),cons_P6512896166579812791at_nat(Y,Ys)),lex_Pr8571645452597969515at_nat(R))
    <=> ( ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),R)
          & ( size_s5460976970255530739at_nat(Xs) = size_s5460976970255530739at_nat(Ys) ) )
        | ( ( X = Y )
          & member6693912407220327184at_nat(produc5943733680697469783at_nat(Xs,Ys),lex_Pr8571645452597969515at_nat(R)) ) ) ) ).

% Cons_in_lex
tff(fact_2943_Cons__in__lex,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,R: set_Pr4329608150637261639at_nat] :
      ( member4080735728053443344at_nat(produc7536900900485677911at_nat(cons_s6881495754146722583at_nat(X,Xs),cons_s6881495754146722583at_nat(Y,Ys)),lex_se2245640040323279819at_nat(R))
    <=> ( ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),R)
          & ( size_s8736152011456118867at_nat(Xs) = size_s8736152011456118867at_nat(Ys) ) )
        | ( ( X = Y )
          & member4080735728053443344at_nat(produc7536900900485677911at_nat(Xs,Ys),lex_se2245640040323279819at_nat(R)) ) ) ) ).

% Cons_in_lex
tff(fact_2944_Cons__in__lex,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: vEBT_VEBT,Ys: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(cons_VEBT_VEBT(X,Xs),cons_VEBT_VEBT(Y,Ys)),lex_VEBT_VEBT(R))
    <=> ( ( member568628332442017744T_VEBT(produc537772716801021591T_VEBT(X,Y),R)
          & ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) )
        | ( ( X = Y )
          & member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Xs,Ys),lex_VEBT_VEBT(R)) ) ) ) ).

% Cons_in_lex
tff(fact_2945_Cons__in__lex,axiom,
    ! [X: bool,Xs: list_bool,Y: bool,Ys: list_bool,R: set_Pr8621012493645324487l_bool] :
      ( member6300827366518646928t_bool(produc970142415094439895t_bool(cons_bool(X,Xs),cons_bool(Y,Ys)),lex_bool(R))
    <=> ( ( member4869097156692242192l_bool(produc8297744024737662807l_bool(X,Y),R)
          & ( size_size_list_bool(Xs) = size_size_list_bool(Ys) ) )
        | ( ( pp(X)
          <=> pp(Y) )
          & member6300827366518646928t_bool(produc970142415094439895t_bool(Xs,Ys),lex_bool(R)) ) ) ) ).

% Cons_in_lex
tff(fact_2946_Cons__in__lex,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(X,Xs),cons_nat(Y,Ys)),lex_nat(R))
    <=> ( ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R)
          & ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) ) )
        | ( ( X = Y )
          & member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),lex_nat(R)) ) ) ) ).

% Cons_in_lex
tff(fact_2947_Cons__in__lex,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(cons_int(X,Xs),cons_int(Y,Ys)),lex_int(R))
    <=> ( ( member5262025264175285858nt_int(product_Pair_int_int(X,Y),R)
          & ( size_size_list_int(Xs) = size_size_list_int(Ys) ) )
        | ( ( X = Y )
          & member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),lex_int(R)) ) ) ) ).

% Cons_in_lex
tff(fact_2948_realpow__pos__nth__unique,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ? [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X3),N) = A )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4))
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y4),N) = A ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_2949_realpow__pos__nth,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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(aa_real_fun_nat_real(power_power_real,R3),N) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_2950_sprop1,axiom,
    ( ( sa = vEBT_Node(info,deg,treeList,summary) )
    & ( deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) )
    & ( size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
    & pp(aa_nat_bool(vEBT_invar_vebt(summary),m))
    & ! [X5: vEBT_VEBT] :
        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(treeList)))
       => pp(aa_nat_bool(vEBT_invar_vebt(X5),na)) ) ) ).

% sprop1
tff(fact_2951_listrel__iff__nth,axiom,
    ! [Xs: list_Code_integer,Ys: list_Code_integer,R: set_Pr4811707699266497531nteger] :
      ( member749217712838834276nteger(produc750622340256944499nteger(Xs,Ys),listre5734910445319291053nteger(R))
    <=> ( ( size_s3445333598471063425nteger(Xs) = size_s3445333598471063425nteger(Ys) )
        & ! [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N4),size_s3445333598471063425nteger(Xs)))
           => member157494554546826820nteger(produc1086072967326762835nteger(aa_nat_Code_integer(nth_Code_integer(Xs),N4),aa_nat_Code_integer(nth_Code_integer(Ys),N4)),R) ) ) ) ).

% listrel_iff_nth
tff(fact_2952_listrel__iff__nth,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Xs,Ys),listre1230615542750757617T_VEBT(R))
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N4),size_s6755466524823107622T_VEBT(Xs)))
           => member568628332442017744T_VEBT(produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,N4),nth_VEBT_VEBT(Ys,N4)),R) ) ) ) ).

% listrel_iff_nth
tff(fact_2953_listrel__iff__nth,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_bool,R: set_Pr2229214029182756970T_bool] :
      ( member6737047059505487539t_bool(produc1275941116658105850t_bool(Xs,Ys),listre7233207804915621076T_bool(R))
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Ys) )
        & ! [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N4),size_s6755466524823107622T_VEBT(Xs)))
           => member3504487111852181555T_bool(produc3035007008238225530T_bool(nth_VEBT_VEBT(Xs,N4),aa_nat_bool(nth_bool(Ys),N4)),R) ) ) ) ).

% listrel_iff_nth
tff(fact_2954_listrel__iff__nth,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,R: set_Pr7556676689462069481BT_nat] :
      ( member6193324644334088288st_nat(produc5570133714943300547st_nat(Xs,Ys),listre5900670229112895443BT_nat(R))
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
        & ! [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N4),size_s6755466524823107622T_VEBT(Xs)))
           => member373505688050248522BT_nat(produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,N4),aa_nat_nat(nth_nat(Ys),N4)),R) ) ) ) ).

% listrel_iff_nth
tff(fact_2955_listrel__iff__nth,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,R: set_Pr5066593544530342725BT_int] :
      ( member3703241499402361532st_int(produc1392282695434103839st_int(Xs,Ys),listre5898179758603845167BT_int(R))
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) )
        & ! [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N4),size_s6755466524823107622T_VEBT(Xs)))
           => member5419026705395827622BT_int(produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,N4),aa_nat_int(nth_int(Ys),N4)),R) ) ) ) ).

% listrel_iff_nth
tff(fact_2956_listrel__iff__nth,axiom,
    ! [Xs: list_bool,Ys: list_VEBT_VEBT,R: set_Pr8097748915547837924T_VEBT] :
      ( member6981531815617313197T_VEBT(produc6468702466802804596T_VEBT(Xs,Ys),listre4049385574073087438T_VEBT(R))
    <=> ( ( size_size_list_bool(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N4),size_size_list_bool(Xs)))
           => member5465154751058088621T_VEBT(produc9074556814250467700T_VEBT(aa_nat_bool(nth_bool(Xs),N4),nth_VEBT_VEBT(Ys,N4)),R) ) ) ) ).

% listrel_iff_nth
tff(fact_2957_listrel__iff__nth,axiom,
    ! [Xs: list_bool,Ys: list_bool,R: set_Pr8621012493645324487l_bool] :
      ( member6300827366518646928t_bool(produc970142415094439895t_bool(Xs,Ys),listrel_bool_bool(R))
    <=> ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
        & ! [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N4),size_size_list_bool(Xs)))
           => member4869097156692242192l_bool(produc8297744024737662807l_bool(aa_nat_bool(nth_bool(Xs),N4),aa_nat_bool(nth_bool(Ys),N4)),R) ) ) ) ).

% listrel_iff_nth
tff(fact_2958_listrel__iff__nth,axiom,
    ! [Xs: list_bool,Ys: list_nat,R: set_Pr7141997262790060364ol_nat] :
      ( member4580749908280483651st_nat(produc2143650348781431462st_nat(Xs,Ys),listrel_bool_nat(R))
    <=> ( ( size_size_list_bool(Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
        & ! [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N4),size_size_list_bool(Xs)))
           => member2564532549135826221ol_nat(produc8874003862241262736ol_nat(aa_nat_bool(nth_bool(Xs),N4),aa_nat_nat(nth_nat(Ys),N4)),R) ) ) ) ).

% listrel_iff_nth
tff(fact_2959_listrel__iff__nth,axiom,
    ! [Xs: list_bool,Ys: list_int,R: set_Pr4651914117858333608ol_int] :
      ( member2090666763348756895st_int(produc7189171366127010562st_int(Xs,Ys),listrel_bool_int(R))
    <=> ( ( size_size_list_bool(Xs) = size_size_list_int(Ys) )
        & ! [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N4),size_size_list_bool(Xs)))
           => member7610053566481405321ol_int(produc8871513391732212460ol_int(aa_nat_bool(nth_bool(Xs),N4),aa_nat_int(nth_int(Ys),N4)),R) ) ) ) ).

% listrel_iff_nth
tff(fact_2960_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_VEBT_VEBT,R: set_Pr6167073792073659919T_VEBT] :
      ( member5968030670617646438T_VEBT(produc8335345208264861441T_VEBT(Xs,Ys),listre5761932458788874033T_VEBT(R))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N4),aa_list_nat_nat(size_size_list_nat,Xs)))
           => member8549952807677709168T_VEBT(produc599794634098209291T_VEBT(aa_nat_nat(nth_nat(Xs),N4),nth_VEBT_VEBT(Ys,N4)),R) ) ) ) ).

% listrel_iff_nth
tff(fact_2961_in__measures_I2_J,axiom,
    ! [X: code_integer,Y: code_integer,F2: fun_Code_integer_nat,Fs: list_f8684324736568799034er_nat] :
      ( member157494554546826820nteger(produc1086072967326762835nteger(X,Y),measur8870801148506250077nteger(cons_f3120178438656451060er_nat(F2,Fs)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(F2,X)),aa_Code_integer_nat(F2,Y)))
        | ( ( aa_Code_integer_nat(F2,X) = aa_Code_integer_nat(F2,Y) )
          & member157494554546826820nteger(produc1086072967326762835nteger(X,Y),measur8870801148506250077nteger(Fs)) ) ) ) ).

% in_measures(2)
tff(fact_2962_in__measures_I2_J,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,F2: fun_Pr2602117396471556196at_nat,Fs: list_f8256639991604547178at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),measur2679027848233739777at_nat(cons_f462676766793977114at_nat(F2,Fs)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Pro4243173770044397973at_nat(F2,X)),aa_Pro4243173770044397973at_nat(F2,Y)))
        | ( ( aa_Pro4243173770044397973at_nat(F2,X) = aa_Pro4243173770044397973at_nat(F2,Y) )
          & member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),measur2679027848233739777at_nat(Fs)) ) ) ) ).

% in_measures(2)
tff(fact_2963_in__measures_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,F2: fun_se4097146774670233028at_nat,Fs: list_f2430256441796990922at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),measur2694323259624372065at_nat(cons_f718425424816052858at_nat(F2,Fs)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set4749074711808745205at_nat(F2,X)),aa_set4749074711808745205at_nat(F2,Y)))
        | ( ( aa_set4749074711808745205at_nat(F2,X) = aa_set4749074711808745205at_nat(F2,Y) )
          & member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),measur2694323259624372065at_nat(Fs)) ) ) ) ).

% in_measures(2)
tff(fact_2964_in__measures_I2_J,axiom,
    ! [X: nat,Y: nat,F2: fun_nat_nat,Fs: list_fun_nat_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),measures_nat(cons_fun_nat_nat(F2,Fs)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F2,X)),aa_nat_nat(F2,Y)))
        | ( ( aa_nat_nat(F2,X) = aa_nat_nat(F2,Y) )
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),measures_nat(Fs)) ) ) ) ).

% in_measures(2)
tff(fact_2965_in__measures_I2_J,axiom,
    ! [X: int,Y: int,F2: fun_int_nat,Fs: list_fun_int_nat] :
      ( member5262025264175285858nt_int(product_Pair_int_int(X,Y),measures_int(cons_fun_int_nat(F2,Fs)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F2,X)),aa_int_nat(F2,Y)))
        | ( ( aa_int_nat(F2,X) = aa_int_nat(F2,Y) )
          & member5262025264175285858nt_int(product_Pair_int_int(X,Y),measures_int(Fs)) ) ) ) ).

% in_measures(2)
tff(fact_2966_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: 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)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_2967_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_2968_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: 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)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_2969_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: 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)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_2970__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062treeList_H_Asummary_H_Ainfo_O_As_A_061_ANode_Ainfo_Adeg_AtreeList_H_Asummary_H_A_092_060and_062_Adeg_A_061_An_A_L_Am_A_092_060and_062_Alength_AtreeList_H_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_H_Am_A_092_060and_062_A_I_092_060forall_062t_092_060in_062set_AtreeList_H_O_Ainvar__vebt_At_An_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,Info: option4927543243414619207at_nat] :
        ~ ( ( sa = vEBT_Node(Info,deg,TreeList3,Summary3) )
          & ( deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) )
          & ( 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))),m) )
          & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),m))
          & ! [X5: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList3)))
             => pp(aa_nat_bool(vEBT_invar_vebt(X5),na)) ) ) ).

% \<open>\<And>thesis. (\<And>treeList' summary' info. s = Node info deg treeList' summary' \<and> deg = n + m \<and> length treeList' = 2 ^ m \<and> invar_vebt summary' m \<and> (\<forall>t\<in>set treeList'. invar_vebt t n) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_2971_intind,axiom,
    ! [I2: nat,N: nat,P2: fun_nat_bool,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
     => ( pp(aa_nat_bool(P2,X))
       => pp(aa_nat_bool(P2,aa_nat_nat(nth_nat(replicate_nat(N,X)),I2))) ) ) ).

% intind
tff(fact_2972_intind,axiom,
    ! [I2: nat,N: nat,P2: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
     => ( pp(aa_VEBT_VEBT_bool(P2,X))
       => pp(aa_VEBT_VEBT_bool(P2,nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X),I2))) ) ) ).

% intind
tff(fact_2973_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( numera1916890842035813515d_enat(M2) = numera1916890842035813515d_enat(N) )
    <=> ( M2 = N ) ) ).

% numeral_eq_iff
tff(fact_2974_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( numera6690914467698888265omplex(M2) = numera6690914467698888265omplex(N) )
    <=> ( M2 = N ) ) ).

% numeral_eq_iff
tff(fact_2975_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( aa_num_real(numeral_numeral_real,M2) = aa_num_real(numeral_numeral_real,N) )
    <=> ( M2 = N ) ) ).

% numeral_eq_iff
tff(fact_2976_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( aa_num_nat(numeral_numeral_nat,M2) = aa_num_nat(numeral_numeral_nat,N) )
    <=> ( M2 = N ) ) ).

% numeral_eq_iff
tff(fact_2977_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( aa_num_int(numeral_numeral_int,M2) = aa_num_int(numeral_numeral_int,N) )
    <=> ( M2 = N ) ) ).

% numeral_eq_iff
tff(fact_2978_semiring__norm_I13_J,axiom,
    ! [M2: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,M2)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N))) ).

% semiring_norm(13)
tff(fact_2979_semiring__norm_I11_J,axiom,
    ! [M2: num] : aa_num_num(aa_num_fun_num_num(times_times_num,M2),one) = M2 ).

% semiring_norm(11)
tff(fact_2980_semiring__norm_I12_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,one),N) = N ).

% semiring_norm(12)
tff(fact_2981_case4_I10_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))) ).

% case4(10)
tff(fact_2982_case4_I4_J,axiom,
    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) ).

% case4(4)
tff(fact_2983_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M2) = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( M2 = N ) ) ).

% of_nat_eq_iff
tff(fact_2984_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M2) = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( M2 = N ) ) ).

% of_nat_eq_iff
tff(fact_2985_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M2) = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( M2 = N ) ) ).

% of_nat_eq_iff
tff(fact_2986_a0,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)) ).

% a0
tff(fact_2987_case4_I7_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(treeList2,I4)),X_13))
      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary2),I4)) ) ) ).

% case4(7)
tff(fact_2988_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N))
       => 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))),N))) ) ) ).

% member_bound
tff(fact_2989_valid__pres__insert,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( 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))),N)))
       => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_insert(T,X)),N)) ) ) ).

% valid_pres_insert
tff(fact_2990_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M2),numera1916890842035813515d_enat(N))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N)) ) ).

% numeral_le_iff
tff(fact_2991_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M2)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N)) ) ).

% numeral_le_iff
tff(fact_2992_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N)) ) ).

% numeral_le_iff
tff(fact_2993_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N)) ) ).

% numeral_le_iff
tff(fact_2994_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N)) ) ).

% numeral_le_iff
tff(fact_2995_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N)) ) ).

% numeral_less_iff
tff(fact_2996_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M2),numera1916890842035813515d_enat(N))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N)) ) ).

% numeral_less_iff
tff(fact_2997_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M2)),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N)) ) ).

% numeral_less_iff
tff(fact_2998_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N)) ) ).

% numeral_less_iff
tff(fact_2999_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N)) ) ).

% numeral_less_iff
tff(fact_3000_numeral__times__numeral,axiom,
    ! [M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N)) ).

% numeral_times_numeral
tff(fact_3001_numeral__times__numeral,axiom,
    ! [M2: num,N: num] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(M2),numera1916890842035813515d_enat(N)) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(times_times_num,M2),N)) ).

% numeral_times_numeral
tff(fact_3002_numeral__times__numeral,axiom,
    ! [M2: num,N: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M2)),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M2),N)) ).

% numeral_times_numeral
tff(fact_3003_numeral__times__numeral,axiom,
    ! [M2: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M2)),aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N)) ).

% numeral_times_numeral
tff(fact_3004_numeral__times__numeral,axiom,
    ! [M2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N)) ).

% numeral_times_numeral
tff(fact_3005_numeral__times__numeral,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N)) ).

% numeral_times_numeral
tff(fact_3006_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W2: num,Z2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),Z2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2))),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_3007_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W2: num,Z2: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(V2),times_7803423173614009249d_enat(numera1916890842035813515d_enat(W2),Z2)) = times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2)),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_3008_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W2: num,Z2: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V2)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),Z2)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2))),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_3009_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W2: num,Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V2)),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(aa_num_fun_num_num(times_times_num,V2),W2))),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_3010_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W2: num,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V2)),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(aa_num_fun_num_num(times_times_num,V2),W2))),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_3011_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W2: num,Z2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V2)),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(aa_num_fun_num_num(times_times_num,V2),W2))),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_3012_add__numeral__left,axiom,
    ! [V2: num,W2: num,Z2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,W2)),Z2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V2),W2))),Z2) ).

% add_numeral_left
tff(fact_3013_add__numeral__left,axiom,
    ! [V2: num,W2: num,Z2: extended_enat] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(V2),plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(W2),Z2)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,V2),W2)),Z2) ).

% add_numeral_left
tff(fact_3014_add__numeral__left,axiom,
    ! [V2: num,W2: num,Z2: complex] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(V2)),aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(W2)),Z2)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V2),W2))),Z2) ).

% add_numeral_left
tff(fact_3015_add__numeral__left,axiom,
    ! [V2: num,W2: num,Z2: real] : aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,V2)),aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,W2)),Z2)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,V2),W2))),Z2) ).

% add_numeral_left
tff(fact_3016_add__numeral__left,axiom,
    ! [V2: num,W2: num,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V2)),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(aa_num_fun_num_num(plus_plus_num,V2),W2))),Z2) ).

% add_numeral_left
tff(fact_3017_add__numeral__left,axiom,
    ! [V2: num,W2: num,Z2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V2)),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(aa_num_fun_num_num(plus_plus_num,V2),W2))),Z2) ).

% add_numeral_left
tff(fact_3018_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)) ).

% numeral_plus_numeral
tff(fact_3019_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(M2),numera1916890842035813515d_enat(N)) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)) ).

% numeral_plus_numeral
tff(fact_3020_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M2)),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)) ).

% numeral_plus_numeral
tff(fact_3021_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] : aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,M2)),aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)) ).

% numeral_plus_numeral
tff(fact_3022_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)) ).

% numeral_plus_numeral
tff(fact_3023_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)) ).

% numeral_plus_numeral
tff(fact_3024_num__double,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,one)),N) = aa_num_num(bit0,N) ).

% num_double
tff(fact_3025_power__mult__numeral,axiom,
    ! [A: nat,M2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M2))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N))) ).

% power_mult_numeral
tff(fact_3026_power__mult__numeral,axiom,
    ! [A: real,M2: num,N: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M2))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N))) ).

% power_mult_numeral
tff(fact_3027_power__mult__numeral,axiom,
    ! [A: int,M2: num,N: num] : aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M2))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N))) ).

% power_mult_numeral
tff(fact_3028_power__mult__numeral,axiom,
    ! [A: complex,M2: num,N: num] : aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,M2))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N))) ).

% power_mult_numeral
tff(fact_3029_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_3030_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( some_nat(M2) = vEBT_vebt_mint(T) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).

% misiz
tff(fact_3031_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_3032_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( 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))),N)))
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,X)),X)) ) ) ).

% valid_insert_both_member_options_add
tff(fact_3033_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( 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))),N)))
       => ( 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))),N)))
         => ( 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_3034_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( 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))),N))) ) ) ).

% helpypredd
tff(fact_3035_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( 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))),N))) ) ) ).

% helpyd
tff(fact_3036_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( 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))),N)))
       => ( 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))),N)))
         => ( 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_3037_replicate__eq__replicate,axiom,
    ! [M2: nat,X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(M2,X) = replicate_VEBT_VEBT(N,Y) )
    <=> ( ( M2 = N )
        & ( ( M2 != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_3038_length__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(N,X)) = N ).

% length_replicate
tff(fact_3039_length__replicate,axiom,
    ! [N: nat,X: bool] : size_size_list_bool(replicate_bool(N,X)) = N ).

% length_replicate
tff(fact_3040_length__replicate,axiom,
    ! [N: nat,X: nat] : aa_list_nat_nat(size_size_list_nat,replicate_nat(N,X)) = N ).

% length_replicate
tff(fact_3041_length__replicate,axiom,
    ! [N: nat,X: int] : size_size_list_int(replicate_int(N,X)) = N ).

% length_replicate
tff(fact_3042_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_3043_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_3044_set__n__deg__not__0,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,M2: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
     => ( ( 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))),M2) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N)) ) ) ).

% set_n_deg_not_0
tff(fact_3045_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: 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)),N))
     => ( 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_3046_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_3047_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_3048_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_3049_distrib__left__numeral,axiom,
    ! [V2: num,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V2)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V2)),C2)) ).

% distrib_left_numeral
tff(fact_3050_distrib__left__numeral,axiom,
    ! [V2: num,B: extended_enat,C2: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(V2),plus_p3455044024723400733d_enat(B,C2)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V2),B),times_7803423173614009249d_enat(numera1916890842035813515d_enat(V2),C2)) ).

% distrib_left_numeral
tff(fact_3051_distrib__left__numeral,axiom,
    ! [V2: num,B: complex,C2: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V2)),aa_complex_complex(plus_plus_complex(B),C2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V2)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V2)),C2)) ).

% distrib_left_numeral
tff(fact_3052_distrib__left__numeral,axiom,
    ! [V2: num,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V2)),aa_real_real(plus_plus_real(B),C2)) = aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V2)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V2)),C2)) ).

% distrib_left_numeral
tff(fact_3053_distrib__left__numeral,axiom,
    ! [V2: num,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) = 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,V2)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V2)),C2)) ).

% distrib_left_numeral
tff(fact_3054_distrib__left__numeral,axiom,
    ! [V2: num,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = 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,V2)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V2)),C2)) ).

% distrib_left_numeral
tff(fact_3055_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V2: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_num_rat(numeral_numeral_rat,V2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,V2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_num_rat(numeral_numeral_rat,V2))) ).

% distrib_right_numeral
tff(fact_3056_distrib__right__numeral,axiom,
    ! [A: extended_enat,B: extended_enat,V2: num] : times_7803423173614009249d_enat(plus_p3455044024723400733d_enat(A,B),numera1916890842035813515d_enat(V2)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(A,numera1916890842035813515d_enat(V2)),times_7803423173614009249d_enat(B,numera1916890842035813515d_enat(V2))) ).

% distrib_right_numeral
tff(fact_3057_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V2: num] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),numera6690914467698888265omplex(V2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V2))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V2))) ).

% distrib_right_numeral
tff(fact_3058_distrib__right__numeral,axiom,
    ! [A: real,B: real,V2: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(plus_plus_real(A),B)),aa_num_real(numeral_numeral_real,V2)) = aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,V2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_num_real(numeral_numeral_real,V2))) ).

% distrib_right_numeral
tff(fact_3059_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V2: 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,V2)) = 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,V2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_num_nat(numeral_numeral_nat,V2))) ).

% distrib_right_numeral
tff(fact_3060_distrib__right__numeral,axiom,
    ! [A: int,B: int,V2: 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,V2)) = 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,V2))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V2))) ).

% distrib_right_numeral
tff(fact_3061_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger(N) = one_one_Code_integer )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_3062_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numera1916890842035813515d_enat(N) = one_on7984719198319812577d_enat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_3063_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex(N) = one_one_complex )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_3064_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_real(numeral_numeral_real,N) = one_one_real )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_3065_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_nat(numeral_numeral_nat,N) = one_one_nat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_3066_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_int(numeral_numeral_int,N) = one_one_int )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_3067_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_Code_integer = numera6620942414471956472nteger(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_3068_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_on7984719198319812577d_enat = numera1916890842035813515d_enat(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_3069_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_3070_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real = aa_num_real(numeral_numeral_real,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_3071_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_3072_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_3073_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V2: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),aa_num_rat(numeral_numeral_rat,V2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,V2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_num_rat(numeral_numeral_rat,V2))) ).

% left_diff_distrib_numeral
tff(fact_3074_left__diff__distrib__numeral,axiom,
    ! [A: complex,B: complex,V2: num] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),numera6690914467698888265omplex(V2)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V2))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V2))) ).

% left_diff_distrib_numeral
tff(fact_3075_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V2: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),B)),aa_num_real(numeral_numeral_real,V2)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,V2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_num_real(numeral_numeral_real,V2))) ).

% left_diff_distrib_numeral
tff(fact_3076_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V2: 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,V2)) = 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,V2))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V2))) ).

% left_diff_distrib_numeral
tff(fact_3077_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V2)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V2)),C2)) ).

% right_diff_distrib_numeral
tff(fact_3078_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: complex,C2: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V2)),aa_complex_complex(minus_minus_complex(B),C2)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V2)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V2)),C2)) ).

% right_diff_distrib_numeral
tff(fact_3079_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V2)),aa_real_real(minus_minus_real(B),C2)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V2)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V2)),C2)) ).

% right_diff_distrib_numeral
tff(fact_3080_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) = 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,V2)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V2)),C2)) ).

% right_diff_distrib_numeral
tff(fact_3081_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_3082_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_3083_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_real ).

% power_zero_numeral
tff(fact_3084_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_int ).

% power_zero_numeral
tff(fact_3085_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_3086_Suc__numeral,axiom,
    ! [N: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% Suc_numeral
tff(fact_3087_power__add__numeral,axiom,
    ! [A: complex,M2: num,N: num] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,M2))),aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N))) ).

% power_add_numeral
tff(fact_3088_power__add__numeral,axiom,
    ! [A: real,M2: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N))) ).

% power_add_numeral
tff(fact_3089_power__add__numeral,axiom,
    ! [A: rat,M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M2))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N))) ).

% power_add_numeral
tff(fact_3090_power__add__numeral,axiom,
    ! [A: nat,M2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N))) ).

% power_add_numeral
tff(fact_3091_power__add__numeral,axiom,
    ! [A: int,M2: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M2))),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N))) ).

% power_add_numeral
tff(fact_3092_power__add__numeral2,axiom,
    ! [A: complex,M2: num,N: num,B: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,M2))),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)))),B) ).

% power_add_numeral2
tff(fact_3093_power__add__numeral2,axiom,
    ! [A: real,M2: num,N: num,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)))),B) ).

% power_add_numeral2
tff(fact_3094_power__add__numeral2,axiom,
    ! [A: rat,M2: num,N: num,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)))),B) ).

% power_add_numeral2
tff(fact_3095_power__add__numeral2,axiom,
    ! [A: nat,M2: num,N: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)))),B) ).

% power_add_numeral2
tff(fact_3096_power__add__numeral2,axiom,
    ! [A: int,M2: num,N: num,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)))),B) ).

% power_add_numeral2
tff(fact_3097_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M2) = zero_zero_complex )
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3098_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M2) = zero_zero_rat )
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3099_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M2) = zero_zero_real )
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3100_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M2) = zero_zero_int )
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3101_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M2) = zero_zero_nat )
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3102_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_3103_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_3104_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_3105_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_3106_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_3107_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_3108_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% 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_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% 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__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% of_nat_less_iff
tff(fact_3113_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% of_nat_less_iff
tff(fact_3114_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% of_nat_less_iff
tff(fact_3115_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% of_nat_less_iff
tff(fact_3116_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_num_nat(numeral_numeral_nat,N)) = numera1916890842035813515d_enat(N) ).

% of_nat_numeral
tff(fact_3117_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,N)) = numera6690914467698888265omplex(N) ).

% of_nat_numeral
tff(fact_3118_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,N)) = aa_num_real(numeral_numeral_real,N) ).

% of_nat_numeral
tff(fact_3119_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).

% of_nat_numeral
tff(fact_3120_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,N) ).

% of_nat_numeral
tff(fact_3121_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% of_nat_le_iff
tff(fact_3122_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% of_nat_le_iff
tff(fact_3123_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% of_nat_le_iff
tff(fact_3124_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% of_nat_le_iff
tff(fact_3125_of__nat__add,axiom,
    ! [M2: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_add
tff(fact_3126_of__nat__add,axiom,
    ! [M2: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_add
tff(fact_3127_of__nat__add,axiom,
    ! [M2: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_add
tff(fact_3128_of__nat__add,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_add
tff(fact_3129_of__nat__mult,axiom,
    ! [M2: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_mult
tff(fact_3130_of__nat__mult,axiom,
    ! [M2: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_mult
tff(fact_3131_of__nat__mult,axiom,
    ! [M2: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mult
tff(fact_3132_of__nat__mult,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_mult
tff(fact_3133_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,N) = one_one_Code_integer )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3134_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,N) = one_one_complex )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3135_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,N) = one_one_real )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3136_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,N) = one_one_int )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3137_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,N) = one_one_nat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3138_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3139_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3140_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3141_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3142_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3143_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_3144_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_3145_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_3146_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_3147_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_3148_in__set__replicate,axiom,
    ! [X: complex,N: nat,Y: complex] :
      ( pp(aa_set_complex_bool(member_complex(X),set_complex2(replicate_complex(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_3149_in__set__replicate,axiom,
    ! [X: real,N: nat,Y: real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(replicate_real(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_3150_in__set__replicate,axiom,
    ! [X: set_nat,N: nat,Y: set_nat] :
      ( member_set_nat(X,set_set_nat2(replicate_set_nat(N,Y)))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_3151_in__set__replicate,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(replicate_int(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_3152_in__set__replicate,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(replicate_nat(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_3153_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_3154_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P2: fun_nat_bool] :
      ( ? [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(replicate_nat(N,A))))
          & pp(aa_nat_bool(P2,X4)) )
    <=> ( pp(aa_nat_bool(P2,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_3155_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P2: fun_VEBT_VEBT_bool] :
      ( ? [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A))))
          & pp(aa_VEBT_VEBT_bool(P2,X4)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P2,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_3156_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P2: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(replicate_nat(N,A))))
         => pp(aa_nat_bool(P2,X4)) )
    <=> ( pp(aa_nat_bool(P2,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_3157_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P2: fun_VEBT_VEBT_bool] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A))))
         => pp(aa_VEBT_VEBT_bool(P2,X4)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P2,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_3158_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(N)),one_one_Code_integer))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_3159_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(N),one_on7984719198319812577d_enat)
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_3160_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),one_one_real))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_3161_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),one_one_rat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_3162_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_3163_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),one_one_int))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_3164_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),numera6620942414471956472nteger(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_3165_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_3166_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ord_le72135733267957522d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(N))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_3167_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_3168_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_3169_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_3170_one__plus__numeral,axiom,
    ! [N: num] : plus_p5714425477246183910nteger(one_one_Code_integer,numera6620942414471956472nteger(N)) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_3171_one__plus__numeral,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_3172_one__plus__numeral,axiom,
    ! [N: num] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(N)) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_3173_one__plus__numeral,axiom,
    ! [N: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_3174_one__plus__numeral,axiom,
    ! [N: num] : aa_real_real(plus_plus_real(one_one_real),aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_3175_one__plus__numeral,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_3176_one__plus__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_3177_numeral__plus__one,axiom,
    ! [N: num] : plus_p5714425477246183910nteger(numera6620942414471956472nteger(N),one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_3178_numeral__plus__one,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,N)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_3179_numeral__plus__one,axiom,
    ! [N: num] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_3180_numeral__plus__one,axiom,
    ! [N: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_3181_numeral__plus__one,axiom,
    ! [N: num] : aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,N)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_3182_numeral__plus__one,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_3183_numeral__plus__one,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_3184_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M2)),zero_zero_real))
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3185_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),zero_zero_rat))
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3186_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),zero_zero_nat))
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3187_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M2)),zero_zero_int))
    <=> ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3188_of__nat__Suc,axiom,
    ! [M2: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,M2)) = plus_p5714425477246183910nteger(one_one_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,M2)) ).

% of_nat_Suc
tff(fact_3189_of__nat__Suc,axiom,
    ! [M2: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M2)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,M2)) ).

% of_nat_Suc
tff(fact_3190_of__nat__Suc,axiom,
    ! [M2: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M2)) ).

% of_nat_Suc
tff(fact_3191_of__nat__Suc,axiom,
    ! [M2: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M2)) = aa_real_real(plus_plus_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,M2)) ).

% of_nat_Suc
tff(fact_3192_of__nat__Suc,axiom,
    ! [M2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,M2)) ).

% of_nat_Suc
tff(fact_3193_of__nat__Suc,axiom,
    ! [M2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M2)) ).

% of_nat_Suc
tff(fact_3194_one__add__one,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_3195_one__add__one,axiom,
    aa_rat_rat(aa_rat_fun_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_3196_one__add__one,axiom,
    plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_3197_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_3198_one__add__one,axiom,
    aa_real_real(plus_plus_real(one_one_real),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_3199_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_3200_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_3201_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_3202_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_3203_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_3204_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_3205_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_3206_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_3207_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_3208_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_3209_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_3210_add__2__eq__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc
tff(fact_3211_add__2__eq__Suc_H,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc'
tff(fact_3212_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_3213_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(aa_real_fun_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_3214_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_3215_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_3216_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_3217_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(aa_real_fun_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_3218_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_3219_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_3220_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_3221_set__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_3222_set__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( N != zero_zero_nat )
     => ( set_nat2(replicate_nat(N,X)) = insert_nat(X,bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_3223_set__replicate,axiom,
    ! [N: nat,X: int] :
      ( ( N != zero_zero_nat )
     => ( set_int2(replicate_int(N,X)) = insert_int(X,bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_3224_set__replicate,axiom,
    ! [N: nat,X: real] :
      ( ( N != zero_zero_nat )
     => ( set_real2(replicate_real(N,X)) = insert_real(X,bot_bot_set_real) ) ) ).

% set_replicate
tff(fact_3225_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(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_3226_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_3227_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_3228_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_3229_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_3230_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_3231_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_3232_zero__less__power2,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_3233_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_3234_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_3235_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_3236_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_3237_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_3238_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I2)),N)),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,I2)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3239_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: 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,I2)),N)),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,I2)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3240_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: 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,I2)),N)),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,I2)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3241_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: 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,I2)),N)),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,I2)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3242_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I2)),N)))
    <=> 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,I2)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3243_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: 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,I2)),N)))
    <=> 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,I2)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3244_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: 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,I2)),N)))
    <=> 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,I2)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3245_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: 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,I2)),N)))
    <=> 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,I2)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3246_power2__nat__le__imp__le,axiom,
    ! [M2: nat,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,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% power2_nat_le_imp_le
tff(fact_3247_power2__nat__le__eq__le,axiom,
    ! [M2: nat,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,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% power2_nat_le_eq_le
tff(fact_3248_self__le__ge2__pow,axiom,
    ! [K2: nat,M2: 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))),K2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),M2))) ) ).

% self_le_ge2_pow
tff(fact_3249_card__2__iff_H,axiom,
    ! [S3: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),S3))
          & ? [Xa3: nat] :
              ( pp(aa_set_nat_bool(member_nat(Xa3),S3))
              & ( X4 != Xa3 )
              & ! [Xb2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xb2),S3))
                 => ( ( Xb2 = X4 )
                    | ( Xb2 = Xa3 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_3250_card__2__iff_H,axiom,
    ! [S3: set_complex] :
      ( ( finite_card_complex(S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),S3))
          & ? [Xa3: complex] :
              ( pp(aa_set_complex_bool(member_complex(Xa3),S3))
              & ( X4 != Xa3 )
              & ! [Xb2: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xb2),S3))
                 => ( ( Xb2 = X4 )
                    | ( Xb2 = Xa3 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_3251_card__2__iff_H,axiom,
    ! [S3: set_int] :
      ( ( finite_card_int(S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),S3))
          & ? [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),S3))
              & ( X4 != Xa3 )
              & ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),S3))
                 => ( ( Xb2 = X4 )
                    | ( Xb2 = Xa3 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_3252_card__2__iff_H,axiom,
    ! [S3: set_list_nat] :
      ( ( finite_card_list_nat(S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X4: list_nat] :
          ( member_list_nat(X4,S3)
          & ? [Xa3: list_nat] :
              ( member_list_nat(Xa3,S3)
              & ( X4 != Xa3 )
              & ! [Xb2: list_nat] :
                  ( member_list_nat(Xb2,S3)
                 => ( ( Xb2 = X4 )
                    | ( Xb2 = Xa3 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_3253_card__2__iff_H,axiom,
    ! [S3: set_set_nat] :
      ( ( finite_card_set_nat(S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X4: set_nat] :
          ( member_set_nat(X4,S3)
          & ? [Xa3: set_nat] :
              ( member_set_nat(Xa3,S3)
              & ( X4 != Xa3 )
              & ! [Xb2: set_nat] :
                  ( member_set_nat(Xb2,S3)
                 => ( ( Xb2 = X4 )
                    | ( Xb2 = Xa3 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_3254_reals__Archimedean3,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ! [Y4: real] :
        ? [N3: 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,N3)),X))) ) ).

% reals_Archimedean3
tff(fact_3255_add__One__commute,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N) = aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one) ).

% add_One_commute
tff(fact_3256_numeral__Bit0,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N)) ).

% numeral_Bit0
tff(fact_3257_numeral__Bit0,axiom,
    ! [N: num] : numera1916890842035813515d_enat(aa_num_num(bit0,N)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N),numera1916890842035813515d_enat(N)) ).

% numeral_Bit0
tff(fact_3258_numeral__Bit0,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit0,N)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N)) ).

% numeral_Bit0
tff(fact_3259_numeral__Bit0,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,N)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N)) ).

% numeral_Bit0
tff(fact_3260_numeral__Bit0,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_Bit0
tff(fact_3261_numeral__Bit0,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).

% numeral_Bit0
tff(fact_3262_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% mult_numeral_1_right
tff(fact_3263_mult__numeral__1__right,axiom,
    ! [A: extended_enat] : times_7803423173614009249d_enat(A,numera1916890842035813515d_enat(one)) = A ).

% mult_numeral_1_right
tff(fact_3264_mult__numeral__1__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).

% mult_numeral_1_right
tff(fact_3265_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_3266_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_3267_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_3268_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,one)),A) = A ).

% mult_numeral_1
tff(fact_3269_mult__numeral__1,axiom,
    ! [A: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(one),A) = A ).

% mult_numeral_1
tff(fact_3270_mult__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).

% mult_numeral_1
tff(fact_3271_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_3272_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_3273_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_3274_numeral__One,axiom,
    numera6620942414471956472nteger(one) = one_one_Code_integer ).

% numeral_One
tff(fact_3275_numeral__One,axiom,
    numera1916890842035813515d_enat(one) = one_on7984719198319812577d_enat ).

% numeral_One
tff(fact_3276_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_3277_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_3278_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_3279_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_3280_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_3281_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_3282_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_3283_mult__2,axiom,
    ! [Z2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z2),Z2) ).

% mult_2
tff(fact_3284_mult__2,axiom,
    ! [Z2: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),Z2) = plus_p3455044024723400733d_enat(Z2,Z2) ).

% mult_2
tff(fact_3285_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_3286_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(plus_plus_real(Z2),Z2) ).

% mult_2
tff(fact_3287_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_3288_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_3289_mult__2__right,axiom,
    ! [Z2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z2),Z2) ).

% mult_2_right
tff(fact_3290_mult__2__right,axiom,
    ! [Z2: extended_enat] : times_7803423173614009249d_enat(Z2,numera1916890842035813515d_enat(aa_num_num(bit0,one))) = plus_p3455044024723400733d_enat(Z2,Z2) ).

% mult_2_right
tff(fact_3291_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_3292_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(plus_plus_real(Z2),Z2) ).

% mult_2_right
tff(fact_3293_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_3294_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_3295_left__add__twice,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3296_left__add__twice,axiom,
    ! [A: extended_enat,B: extended_enat] : plus_p3455044024723400733d_enat(A,plus_p3455044024723400733d_enat(A,B)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),A),B) ).

% left_add_twice
tff(fact_3297_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_3298_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3299_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_3300_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_3301_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_3302_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_3303_zero__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).

% zero_power2
tff(fact_3304_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_3305_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_3306_power2__eq__square,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(times_times_complex(A),A) ).

% power2_eq_square
tff(fact_3307_power2__eq__square,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).

% power2_eq_square
tff(fact_3308_power2__eq__square,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) ).

% power2_eq_square
tff(fact_3309_power2__eq__square,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A) ).

% power2_eq_square
tff(fact_3310_power2__eq__square,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% power2_eq_square
tff(fact_3311_power4__eq__xxxx,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_3312_power4__eq__xxxx,axiom,
    ! [X: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_3313_power4__eq__xxxx,axiom,
    ! [X: rat] : aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_3314_power4__eq__xxxx,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_3315_power4__eq__xxxx,axiom,
    ! [X: int] : aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_3316_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_3317_power__even__eq,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_3318_power__even__eq,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_3319_power__even__eq,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_3320_power__even__eq,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_3321_diff__le__diff__pow,axiom,
    ! [K2: nat,M2: nat,N: 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))),K2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),N)))) ) ).

% diff_le_diff_pow
tff(fact_3322_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_3323_power2__sum,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3324_power2__sum,axiom,
    ! [X: extended_enat,Y: extended_enat] : power_8040749407984259932d_enat(plus_p3455044024723400733d_enat(X,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = plus_p3455044024723400733d_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)))),times_7803423173614009249d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),X),Y)) ).

% power2_sum
tff(fact_3325_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_3326_power2__sum,axiom,
    ! [X: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3327_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_3328_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_3329_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(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_3330_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_3331_power2__diff,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_3332_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_3333_power2__diff,axiom,
    ! [X: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_3334_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_3335_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( 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_3336_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_3337_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_3338_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_3339_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( 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_3340_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_3341_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_3342_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_3343_zero__le__power2,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_3344_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_3345_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_3346_power2__less__0,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)) ).

% power2_less_0
tff(fact_3347_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_3348_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_3349_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_3350_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_3351_card__2__iff,axiom,
    ! [S3: set_complex] :
      ( ( finite_card_complex(S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X4: complex,Y5: complex] :
          ( ( S3 = insert_complex(X4,insert_complex(Y5,bot_bot_set_complex)) )
          & ( X4 != Y5 ) ) ) ).

% card_2_iff
tff(fact_3352_card__2__iff,axiom,
    ! [S3: set_list_nat] :
      ( ( finite_card_list_nat(S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X4: list_nat,Y5: list_nat] :
          ( ( S3 = insert_list_nat(X4,insert_list_nat(Y5,bot_bot_set_list_nat)) )
          & ( X4 != Y5 ) ) ) ).

% card_2_iff
tff(fact_3353_card__2__iff,axiom,
    ! [S3: set_set_nat] :
      ( ( finite_card_set_nat(S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X4: set_nat,Y5: set_nat] :
          ( ( S3 = insert_set_nat(X4,insert_set_nat(Y5,bot_bot_set_set_nat)) )
          & ( X4 != Y5 ) ) ) ).

% card_2_iff
tff(fact_3354_card__2__iff,axiom,
    ! [S3: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X4: nat,Y5: nat] :
          ( ( S3 = insert_nat(X4,insert_nat(Y5,bot_bot_set_nat)) )
          & ( X4 != Y5 ) ) ) ).

% card_2_iff
tff(fact_3355_card__2__iff,axiom,
    ! [S3: set_int] :
      ( ( finite_card_int(S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X4: int,Y5: int] :
          ( ( S3 = insert_int(X4,insert_int(Y5,bot_bot_set_int)) )
          & ( X4 != Y5 ) ) ) ).

% card_2_iff
tff(fact_3356_card__2__iff,axiom,
    ! [S3: set_real] :
      ( ( finite_card_real(S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ? [X4: real,Y5: real] :
          ( ( S3 = insert_real(X4,insert_real(Y5,bot_bot_set_real)) )
          & ( X4 != Y5 ) ) ) ).

% card_2_iff
tff(fact_3357_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_3358_Suc__nat__number__of__add,axiom,
    ! [V2: num,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V2)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V2),one))),N) ).

% Suc_nat_number_of_add
tff(fact_3359_real__arch__simple,axiom,
    ! [X: real] :
    ? [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N3))) ).

% real_arch_simple
tff(fact_3360_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N3))) ).

% real_arch_simple
tff(fact_3361_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N3))) ).

% reals_Archimedean2
tff(fact_3362_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,N3))) ).

% reals_Archimedean2
tff(fact_3363_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(semiri681578069525770553at_rat,X)) ).

% mult_of_nat_commute
tff(fact_3364_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_3365_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_3366_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_3367_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_3368_int__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ).

% int_Suc
tff(fact_3369_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_3370_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,N) ).

% zero_neq_numeral
tff(fact_3371_zero__neq__numeral,axiom,
    ! [N: num] : zero_z5237406670263579293d_enat != numera1916890842035813515d_enat(N) ).

% zero_neq_numeral
tff(fact_3372_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_complex != numera6690914467698888265omplex(N) ).

% zero_neq_numeral
tff(fact_3373_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_num_real(numeral_numeral_real,N) ).

% zero_neq_numeral
tff(fact_3374_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,N) ).

% zero_neq_numeral
tff(fact_3375_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_num_int(numeral_numeral_int,N) ).

% zero_neq_numeral
tff(fact_3376_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_3377_nat__le__real__less,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,M2)),one_one_real))) ) ).

% nat_le_real_less
tff(fact_3378_zle__int,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% zle_int
tff(fact_3379_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_3380_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_3381_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( 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_3382_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_3383_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_3384_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_3385_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real))
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3386_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat))
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3387_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_3388_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(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_3389_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_3390_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_3391_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(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3392_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3393_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_3394_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)) ).

% not_sum_power2_lt_zero
tff(fact_3395_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)) ).

% not_sum_power2_lt_zero
tff(fact_3396_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_3397_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_3398_zero__le__even__power_H,axiom,
    ! [A: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% zero_le_even_power'
tff(fact_3399_zero__le__even__power_H,axiom,
    ! [A: rat,N: 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))),N)))) ).

% zero_le_even_power'
tff(fact_3400_zero__le__even__power_H,axiom,
    ! [A: int,N: 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))),N)))) ).

% zero_le_even_power'
tff(fact_3401_power__odd__eq,axiom,
    ! [A: complex,N: 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))),N))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3402_power__odd__eq,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3403_power__odd__eq,axiom,
    ! [A: rat,N: 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))),N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3404_power__odd__eq,axiom,
    ! [A: nat,N: 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))),N))) = 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),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3405_power__odd__eq,axiom,
    ! [A: int,N: 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))),N))) = 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),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3406_ex__power__ivl1,axiom,
    ! [B: nat,K2: 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),K2))
       => ? [N3: 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),N3)),K2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),one_one_nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_3407_ex__power__ivl2,axiom,
    ! [B: nat,K2: 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))),K2))
       => ? [N3: 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),N3)),K2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),one_one_nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_3408_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
    ! [Deg: nat,X: nat,Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( 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))),Deg)),X))
       => ( vEBT_VEBT_insert(vEBT_Node(Info2,Deg,TreeList2,Summary),X) = vEBT_Node(Info2,Deg,TreeList2,Summary) ) )
      & ( ~ 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))),Deg)),X))
       => ( vEBT_VEBT_insert(vEBT_Node(Info2,Deg,TreeList2,Summary),X) = vEBT_vebt_insert(vEBT_Node(Info2,Deg,TreeList2,Summary),X) ) ) ) ).

% VEBT_internal.insert'.simps(2)
tff(fact_3409_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3410_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N: 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))),N)))))
     => 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_3411_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N: 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))),N)))))
     => 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_3412_odd__power__less__zero,axiom,
    ! [A: real,N: 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(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_real)) ) ).

% odd_power_less_zero
tff(fact_3413_odd__power__less__zero,axiom,
    ! [A: rat,N: 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))),N)))),zero_zero_rat)) ) ).

% odd_power_less_zero
tff(fact_3414_odd__power__less__zero,axiom,
    ! [A: int,N: 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))),N)))),zero_zero_int)) ) ).

% odd_power_less_zero
tff(fact_3415_VEBT__internal_Oinsert_H_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) )
           => ( Y != vEBT_vebt_insert(vEBT_Leaf(A3,B3),Xa2) ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
             => ~ ( ( 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))),Deg2)),Xa2))
                   => ( Y = vEBT_Node(Info,Deg2,TreeList,Summary2) ) )
                  & ( ~ 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))),Deg2)),Xa2))
                   => ( Y = vEBT_vebt_insert(vEBT_Node(Info,Deg2,TreeList,Summary2),Xa2) ) ) ) ) ) ) ).

% VEBT_internal.insert'.elims
tff(fact_3416_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% of_nat_0_le_iff
tff(fact_3417_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% of_nat_0_le_iff
tff(fact_3418_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).

% of_nat_0_le_iff
tff(fact_3419_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% of_nat_0_le_iff
tff(fact_3420_of__nat__less__0__iff,axiom,
    ! [M2: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),zero_zero_rat)) ).

% of_nat_less_0_iff
tff(fact_3421_of__nat__less__0__iff,axiom,
    ! [M2: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M2)),zero_zero_real)) ).

% of_nat_less_0_iff
tff(fact_3422_of__nat__less__0__iff,axiom,
    ! [M2: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M2)),zero_zero_int)) ).

% of_nat_less_0_iff
tff(fact_3423_of__nat__less__0__iff,axiom,
    ! [M2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),zero_zero_nat)) ).

% of_nat_less_0_iff
tff(fact_3424_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,N)) != zero_zero_complex ).

% of_nat_neq_0
tff(fact_3425_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_3426_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_3427_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_3428_of__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_3429_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% of_nat_less_imp_less
tff(fact_3430_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% of_nat_less_imp_less
tff(fact_3431_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% of_nat_less_imp_less
tff(fact_3432_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% of_nat_less_imp_less
tff(fact_3433_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,N))) ) ).

% less_imp_of_nat_less
tff(fact_3434_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,N))) ) ).

% less_imp_of_nat_less
tff(fact_3435_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% less_imp_of_nat_less
tff(fact_3436_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N))) ) ).

% less_imp_of_nat_less
tff(fact_3437_of__nat__mono,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I2)),aa_nat_real(semiri5074537144036343181t_real,J2))) ) ).

% of_nat_mono
tff(fact_3438_of__nat__mono,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I2)),aa_nat_rat(semiri681578069525770553at_rat,J2))) ) ).

% of_nat_mono
tff(fact_3439_of__nat__mono,axiom,
    ! [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_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I2)),aa_nat_nat(semiri1316708129612266289at_nat,J2))) ) ).

% of_nat_mono
tff(fact_3440_of__nat__mono,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I2)),aa_nat_int(semiri1314217659103216013at_int,J2))) ) ).

% of_nat_mono
tff(fact_3441_zero__le__numeral,axiom,
    ! [N: num] : ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat,numera1916890842035813515d_enat(N)) ).

% zero_le_numeral
tff(fact_3442_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,N))) ).

% zero_le_numeral
tff(fact_3443_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% zero_le_numeral
tff(fact_3444_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% zero_le_numeral
tff(fact_3445_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,N))) ).

% zero_le_numeral
tff(fact_3446_not__numeral__le__zero,axiom,
    ! [N: num] : ~ ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(N),zero_z5237406670263579293d_enat) ).

% not_numeral_le_zero
tff(fact_3447_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),zero_zero_real)) ).

% not_numeral_le_zero
tff(fact_3448_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N)),zero_zero_rat)) ).

% not_numeral_le_zero
tff(fact_3449_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).

% not_numeral_le_zero
tff(fact_3450_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int)) ).

% not_numeral_le_zero
tff(fact_3451_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),zero_zero_rat)) ).

% not_numeral_less_zero
tff(fact_3452_not__numeral__less__zero,axiom,
    ! [N: num] : ~ ord_le72135733267957522d_enat(numera1916890842035813515d_enat(N),zero_z5237406670263579293d_enat) ).

% not_numeral_less_zero
tff(fact_3453_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N)),zero_zero_real)) ).

% not_numeral_less_zero
tff(fact_3454_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).

% not_numeral_less_zero
tff(fact_3455_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int)) ).

% not_numeral_less_zero
tff(fact_3456_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% zero_less_numeral
tff(fact_3457_zero__less__numeral,axiom,
    ! [N: num] : ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,numera1916890842035813515d_enat(N)) ).

% zero_less_numeral
tff(fact_3458_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,N))) ).

% zero_less_numeral
tff(fact_3459_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% zero_less_numeral
tff(fact_3460_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,N))) ).

% zero_less_numeral
tff(fact_3461_replicate__Suc,axiom,
    ! [N: nat,X: nat] : replicate_nat(aa_nat_nat(suc,N),X) = cons_nat(X,replicate_nat(N,X)) ).

% replicate_Suc
tff(fact_3462_replicate__Suc,axiom,
    ! [N: nat,X: vEBT_VEBT] : replicate_VEBT_VEBT(aa_nat_nat(suc,N),X) = cons_VEBT_VEBT(X,replicate_VEBT_VEBT(N,X)) ).

% replicate_Suc
tff(fact_3463_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C2: 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),C2))
       => ( ! [M: nat] :
              ( 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(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),X)),C2)) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_3464_one__le__numeral,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),numera6620942414471956472nteger(N))) ).

% one_le_numeral
tff(fact_3465_one__le__numeral,axiom,
    ! [N: num] : ord_le2932123472753598470d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(N)) ).

% one_le_numeral
tff(fact_3466_one__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,N))) ).

% one_le_numeral
tff(fact_3467_one__le__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N))) ).

% one_le_numeral
tff(fact_3468_one__le__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% one_le_numeral
tff(fact_3469_one__le__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,N))) ).

% one_le_numeral
tff(fact_3470_pos__int__cases,axiom,
    ! [K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
     => ~ ! [N3: nat] :
            ( ( K2 = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3)) ) ) ).

% pos_int_cases
tff(fact_3471_zero__less__imp__eq__int,axiom,
    ! [K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
     => ? [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
          & ( K2 = aa_nat_int(semiri1314217659103216013at_int,N3) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_3472_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J2: int,K2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),J2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
       => 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,K2)),I2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K2)),J2))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_3473_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(N)),one_one_Code_integer)) ).

% not_numeral_less_one
tff(fact_3474_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N)),one_one_rat)) ).

% not_numeral_less_one
tff(fact_3475_not__numeral__less__one,axiom,
    ! [N: num] : ~ ord_le72135733267957522d_enat(numera1916890842035813515d_enat(N),one_on7984719198319812577d_enat) ).

% not_numeral_less_one
tff(fact_3476_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N)),one_one_real)) ).

% not_numeral_less_one
tff(fact_3477_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat)) ).

% not_numeral_less_one
tff(fact_3478_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),one_one_int)) ).

% not_numeral_less_one
tff(fact_3479_one__plus__numeral__commute,axiom,
    ! [X: num] : plus_p5714425477246183910nteger(one_one_Code_integer,numera6620942414471956472nteger(X)) = plus_p5714425477246183910nteger(numera6620942414471956472nteger(X),one_one_Code_integer) ).

% one_plus_numeral_commute
tff(fact_3480_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,X)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_3481_one__plus__numeral__commute,axiom,
    ! [X: num] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(X)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(X),one_on7984719198319812577d_enat) ).

% one_plus_numeral_commute
tff(fact_3482_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_3483_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_real_real(plus_plus_real(one_one_real),aa_num_real(numeral_numeral_real,X)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,X)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_3484_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_3485_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_3486_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
         => ( X3 = X ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_3487_replicate__length__same,axiom,
    ! [Xs: list_bool,X: bool] :
      ( ! [X3: bool] :
          ( member_bool(X3,set_bool2(Xs))
         => ( pp(X3)
          <=> pp(X) ) )
     => ( replicate_bool(size_size_list_bool(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_3488_replicate__length__same,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
         => ( X3 = X ) )
     => ( replicate_nat(aa_list_nat_nat(size_size_list_nat,Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_3489_replicate__length__same,axiom,
    ! [Xs: list_int,X: int] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
         => ( X3 = X ) )
     => ( replicate_int(size_size_list_int(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_3490_replicate__eqI,axiom,
    ! [Xs: list_complex,N: nat,X: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = N )
     => ( ! [Y3: complex] :
            ( pp(aa_set_complex_bool(member_complex(Y3),set_complex2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_complex(N,X) ) ) ) ).

% replicate_eqI
tff(fact_3491_replicate__eqI,axiom,
    ! [Xs: list_real,N: nat,X: real] :
      ( ( size_size_list_real(Xs) = N )
     => ( ! [Y3: real] :
            ( pp(aa_set_real_bool(member_real(Y3),set_real2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_real(N,X) ) ) ) ).

% replicate_eqI
tff(fact_3492_replicate__eqI,axiom,
    ! [Xs: list_set_nat,N: nat,X: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = N )
     => ( ! [Y3: set_nat] :
            ( member_set_nat(Y3,set_set_nat2(Xs))
           => ( Y3 = X ) )
       => ( Xs = replicate_set_nat(N,X) ) ) ) ).

% replicate_eqI
tff(fact_3493_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = N )
     => ( ! [Y3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y3),set_VEBT_VEBT2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_VEBT_VEBT(N,X) ) ) ) ).

% replicate_eqI
tff(fact_3494_replicate__eqI,axiom,
    ! [Xs: list_bool,N: nat,X: bool] :
      ( ( size_size_list_bool(Xs) = N )
     => ( ! [Y3: bool] :
            ( member_bool(Y3,set_bool2(Xs))
           => ( pp(Y3)
            <=> pp(X) ) )
       => ( Xs = replicate_bool(N,X) ) ) ) ).

% replicate_eqI
tff(fact_3495_replicate__eqI,axiom,
    ! [Xs: list_nat,N: nat,X: nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = N )
     => ( ! [Y3: nat] :
            ( pp(aa_set_nat_bool(member_nat(Y3),set_nat2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_nat(N,X) ) ) ) ).

% replicate_eqI
tff(fact_3496_replicate__eqI,axiom,
    ! [Xs: list_int,N: nat,X: int] :
      ( ( size_size_list_int(Xs) = N )
     => ( ! [Y3: int] :
            ( pp(aa_set_int_bool(member_int(Y3),set_int2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_int(N,X) ) ) ) ).

% replicate_eqI
tff(fact_3497_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M2))
       => ( ( 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))),M2) )
         => ( ( M2 = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
             => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12))
               => ( ! [X3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                     => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_3498_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M2))
       => ( ( 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))),M2) )
         => ( ( M2 = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
             => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12))
               => ( ! [X3: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                     => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_3499_listrel__eq__len,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Xs,Ys),listre1230615542750757617T_VEBT(R))
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% listrel_eq_len
tff(fact_3500_listrel__eq__len,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_bool,R: set_Pr2229214029182756970T_bool] :
      ( member6737047059505487539t_bool(produc1275941116658105850t_bool(Xs,Ys),listre7233207804915621076T_bool(R))
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Ys) ) ) ).

% listrel_eq_len
tff(fact_3501_listrel__eq__len,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,R: set_Pr7556676689462069481BT_nat] :
      ( member6193324644334088288st_nat(produc5570133714943300547st_nat(Xs,Ys),listre5900670229112895443BT_nat(R))
     => ( size_s6755466524823107622T_VEBT(Xs) = aa_list_nat_nat(size_size_list_nat,Ys) ) ) ).

% listrel_eq_len
tff(fact_3502_listrel__eq__len,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,R: set_Pr5066593544530342725BT_int] :
      ( member3703241499402361532st_int(produc1392282695434103839st_int(Xs,Ys),listre5898179758603845167BT_int(R))
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) ) ) ).

% listrel_eq_len
tff(fact_3503_listrel__eq__len,axiom,
    ! [Xs: list_bool,Ys: list_VEBT_VEBT,R: set_Pr8097748915547837924T_VEBT] :
      ( member6981531815617313197T_VEBT(produc6468702466802804596T_VEBT(Xs,Ys),listre4049385574073087438T_VEBT(R))
     => ( size_size_list_bool(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% listrel_eq_len
tff(fact_3504_listrel__eq__len,axiom,
    ! [Xs: list_bool,Ys: list_bool,R: set_Pr8621012493645324487l_bool] :
      ( member6300827366518646928t_bool(produc970142415094439895t_bool(Xs,Ys),listrel_bool_bool(R))
     => ( size_size_list_bool(Xs) = size_size_list_bool(Ys) ) ) ).

% listrel_eq_len
tff(fact_3505_listrel__eq__len,axiom,
    ! [Xs: list_bool,Ys: list_nat,R: set_Pr7141997262790060364ol_nat] :
      ( member4580749908280483651st_nat(produc2143650348781431462st_nat(Xs,Ys),listrel_bool_nat(R))
     => ( size_size_list_bool(Xs) = aa_list_nat_nat(size_size_list_nat,Ys) ) ) ).

% listrel_eq_len
tff(fact_3506_listrel__eq__len,axiom,
    ! [Xs: list_bool,Ys: list_int,R: set_Pr4651914117858333608ol_int] :
      ( member2090666763348756895st_int(produc7189171366127010562st_int(Xs,Ys),listrel_bool_int(R))
     => ( size_size_list_bool(Xs) = size_size_list_int(Ys) ) ) ).

% listrel_eq_len
tff(fact_3507_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_VEBT_VEBT,R: set_Pr6167073792073659919T_VEBT] :
      ( member5968030670617646438T_VEBT(produc8335345208264861441T_VEBT(Xs,Ys),listre5761932458788874033T_VEBT(R))
     => ( aa_list_nat_nat(size_size_list_nat,Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% listrel_eq_len
tff(fact_3508_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_bool,R: set_Pr4681767519595310322t_bool] :
      ( member702678106789554249t_bool(produc5531179457150407524t_bool(Xs,Ys),listrel_nat_bool(R))
     => ( aa_list_nat_nat(size_size_list_nat,Xs) = size_size_list_bool(Ys) ) ) ).

% listrel_eq_len
tff(fact_3509_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))
     => ? [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N3)),X))) ) ).

% ex_less_of_nat_mult
tff(fact_3510_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [N3: 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,N3)),X))) ) ).

% ex_less_of_nat_mult
tff(fact_3511_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ) ).

% of_nat_diff
tff(fact_3512_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% of_nat_diff
tff(fact_3513_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ).

% of_nat_diff
tff(fact_3514_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ) ) ).

% of_nat_diff
tff(fact_3515_zdiff__int__split,axiom,
    ! [P2: fun_int_bool,X: nat,Y: nat] :
      ( pp(aa_int_bool(P2,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(P2,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(P2,zero_zero_int)) ) ) ) ).

% zdiff_int_split
tff(fact_3516_num_Osize_I5_J,axiom,
    ! [X2: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_3517_realpow__pos__nth2,axiom,
    ! [A: real,N: 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(aa_real_fun_nat_real(power_power_real,R3),aa_nat_nat(suc,N)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_3518_listrel__Cons2,axiom,
    ! [Xs: list_Code_integer,Y: code_integer,Ys: list_Code_integer,R: set_Pr4811707699266497531nteger] :
      ( member749217712838834276nteger(produc750622340256944499nteger(Xs,cons_Code_integer(Y,Ys)),listre5734910445319291053nteger(R))
     => ~ ! [X3: code_integer,Xs2: list_Code_integer] :
            ( ( Xs = cons_Code_integer(X3,Xs2) )
           => ( member157494554546826820nteger(produc1086072967326762835nteger(X3,Y),R)
             => ~ member749217712838834276nteger(produc750622340256944499nteger(Xs2,Ys),listre5734910445319291053nteger(R)) ) ) ) ).

% listrel_Cons2
tff(fact_3519_listrel__Cons2,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
      ( member6693912407220327184at_nat(produc5943733680697469783at_nat(Xs,cons_P6512896166579812791at_nat(Y,Ys)),listre818007680106770737at_nat(R))
     => ~ ! [X3: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
            ( ( Xs = cons_P6512896166579812791at_nat(X3,Xs2) )
           => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X3,Y),R)
             => ~ member6693912407220327184at_nat(produc5943733680697469783at_nat(Xs2,Ys),listre818007680106770737at_nat(R)) ) ) ) ).

% listrel_Cons2
tff(fact_3520_listrel__Cons2,axiom,
    ! [Xs: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,R: set_Pr4329608150637261639at_nat] :
      ( member4080735728053443344at_nat(produc7536900900485677911at_nat(Xs,cons_s6881495754146722583at_nat(Y,Ys)),listre2047417242196832561at_nat(R))
     => ~ ! [X3: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat] :
            ( ( Xs = cons_s6881495754146722583at_nat(X3,Xs2) )
           => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X3,Y),R)
             => ~ member4080735728053443344at_nat(produc7536900900485677911at_nat(Xs2,Ys),listre2047417242196832561at_nat(R)) ) ) ) ).

% listrel_Cons2
tff(fact_3521_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,cons_nat(Y,Ys)),listrel_nat_nat(R))
     => ~ ! [X3: nat,Xs2: list_nat] :
            ( ( Xs = cons_nat(X3,Xs2) )
           => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y),R)
             => ~ member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs2,Ys),listrel_nat_nat(R)) ) ) ) ).

% listrel_Cons2
tff(fact_3522_listrel__Cons2,axiom,
    ! [Xs: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,cons_int(Y,Ys)),listrel_int_int(R))
     => ~ ! [X3: int,Xs2: list_int] :
            ( ( Xs = cons_int(X3,Xs2) )
           => ( member5262025264175285858nt_int(product_Pair_int_int(X3,Y),R)
             => ~ member6698963635872716290st_int(produc364263696895485585st_int(Xs2,Ys),listrel_int_int(R)) ) ) ) ).

% listrel_Cons2
tff(fact_3523_listrel__Cons1,axiom,
    ! [Y: code_integer,Ys: list_Code_integer,Xs: list_Code_integer,R: set_Pr4811707699266497531nteger] :
      ( member749217712838834276nteger(produc750622340256944499nteger(cons_Code_integer(Y,Ys),Xs),listre5734910445319291053nteger(R))
     => ~ ! [Y3: code_integer,Ys5: list_Code_integer] :
            ( ( Xs = cons_Code_integer(Y3,Ys5) )
           => ( member157494554546826820nteger(produc1086072967326762835nteger(Y,Y3),R)
             => ~ member749217712838834276nteger(produc750622340256944499nteger(Ys,Ys5),listre5734910445319291053nteger(R)) ) ) ) ).

% listrel_Cons1
tff(fact_3524_listrel__Cons1,axiom,
    ! [Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
      ( member6693912407220327184at_nat(produc5943733680697469783at_nat(cons_P6512896166579812791at_nat(Y,Ys),Xs),listre818007680106770737at_nat(R))
     => ~ ! [Y3: product_prod_nat_nat,Ys5: list_P6011104703257516679at_nat] :
            ( ( Xs = cons_P6512896166579812791at_nat(Y3,Ys5) )
           => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(Y,Y3),R)
             => ~ member6693912407220327184at_nat(produc5943733680697469783at_nat(Ys,Ys5),listre818007680106770737at_nat(R)) ) ) ) ).

% listrel_Cons1
tff(fact_3525_listrel__Cons1,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,Xs: list_s1210847774152347623at_nat,R: set_Pr4329608150637261639at_nat] :
      ( member4080735728053443344at_nat(produc7536900900485677911at_nat(cons_s6881495754146722583at_nat(Y,Ys),Xs),listre2047417242196832561at_nat(R))
     => ~ ! [Y3: set_Pr1261947904930325089at_nat,Ys5: list_s1210847774152347623at_nat] :
            ( ( Xs = cons_s6881495754146722583at_nat(Y3,Ys5) )
           => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(Y,Y3),R)
             => ~ member4080735728053443344at_nat(produc7536900900485677911at_nat(Ys,Ys5),listre2047417242196832561at_nat(R)) ) ) ) ).

% listrel_Cons1
tff(fact_3526_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(Y,Ys),Xs),listrel_nat_nat(R))
     => ~ ! [Y3: nat,Ys5: list_nat] :
            ( ( Xs = cons_nat(Y3,Ys5) )
           => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),Y3),R)
             => ~ member7340969449405702474st_nat(produc2694037385005941721st_nat(Ys,Ys5),listrel_nat_nat(R)) ) ) ) ).

% listrel_Cons1
tff(fact_3527_listrel__Cons1,axiom,
    ! [Y: int,Ys: list_int,Xs: list_int,R: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(cons_int(Y,Ys),Xs),listrel_int_int(R))
     => ~ ! [Y3: int,Ys5: list_int] :
            ( ( Xs = cons_int(Y3,Ys5) )
           => ( member5262025264175285858nt_int(product_Pair_int_int(Y,Y3),R)
             => ~ member6698963635872716290st_int(produc364263696895485585st_int(Ys,Ys5),listrel_int_int(R)) ) ) ) ).

% listrel_Cons1
tff(fact_3528_listrel_OCons,axiom,
    ! [X: code_integer,Y: code_integer,R: set_Pr4811707699266497531nteger,Xs: list_Code_integer,Ys: list_Code_integer] :
      ( member157494554546826820nteger(produc1086072967326762835nteger(X,Y),R)
     => ( member749217712838834276nteger(produc750622340256944499nteger(Xs,Ys),listre5734910445319291053nteger(R))
       => member749217712838834276nteger(produc750622340256944499nteger(cons_Code_integer(X,Xs),cons_Code_integer(Y,Ys)),listre5734910445319291053nteger(R)) ) ) ).

% listrel.Cons
tff(fact_3529_listrel_OCons,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),R)
     => ( member6693912407220327184at_nat(produc5943733680697469783at_nat(Xs,Ys),listre818007680106770737at_nat(R))
       => member6693912407220327184at_nat(produc5943733680697469783at_nat(cons_P6512896166579812791at_nat(X,Xs),cons_P6512896166579812791at_nat(Y,Ys)),listre818007680106770737at_nat(R)) ) ) ).

% listrel.Cons
tff(fact_3530_listrel_OCons,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),R)
     => ( member4080735728053443344at_nat(produc7536900900485677911at_nat(Xs,Ys),listre2047417242196832561at_nat(R))
       => member4080735728053443344at_nat(produc7536900900485677911at_nat(cons_s6881495754146722583at_nat(X,Xs),cons_s6881495754146722583at_nat(Y,Ys)),listre2047417242196832561at_nat(R)) ) ) ).

% listrel.Cons
tff(fact_3531_listrel_OCons,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R)
     => ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel_nat_nat(R))
       => member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(X,Xs),cons_nat(Y,Ys)),listrel_nat_nat(R)) ) ) ).

% listrel.Cons
tff(fact_3532_listrel_OCons,axiom,
    ! [X: int,Y: int,R: set_Pr958786334691620121nt_int,Xs: list_int,Ys: list_int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(X,Y),R)
     => ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel_int_int(R))
       => member6698963635872716290st_int(produc364263696895485585st_int(cons_int(X,Xs),cons_int(Y,Ys)),listrel_int_int(R)) ) ) ).

% listrel.Cons
tff(fact_3533_measures__less,axiom,
    ! [F2: fun_Code_integer_nat,X: code_integer,Y: code_integer,Fs: list_f8684324736568799034er_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(F2,X)),aa_Code_integer_nat(F2,Y)))
     => member157494554546826820nteger(produc1086072967326762835nteger(X,Y),measur8870801148506250077nteger(cons_f3120178438656451060er_nat(F2,Fs))) ) ).

% measures_less
tff(fact_3534_measures__less,axiom,
    ! [F2: fun_Pr2602117396471556196at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Fs: list_f8256639991604547178at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Pro4243173770044397973at_nat(F2,X)),aa_Pro4243173770044397973at_nat(F2,Y)))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),measur2679027848233739777at_nat(cons_f462676766793977114at_nat(F2,Fs))) ) ).

% measures_less
tff(fact_3535_measures__less,axiom,
    ! [F2: fun_se4097146774670233028at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Fs: list_f2430256441796990922at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set4749074711808745205at_nat(F2,X)),aa_set4749074711808745205at_nat(F2,Y)))
     => member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),measur2694323259624372065at_nat(cons_f718425424816052858at_nat(F2,Fs))) ) ).

% measures_less
tff(fact_3536_measures__less,axiom,
    ! [F2: fun_nat_nat,X: nat,Y: nat,Fs: list_fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F2,X)),aa_nat_nat(F2,Y)))
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),measures_nat(cons_fun_nat_nat(F2,Fs))) ) ).

% measures_less
tff(fact_3537_measures__less,axiom,
    ! [F2: fun_int_nat,X: int,Y: int,Fs: list_fun_int_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F2,X)),aa_int_nat(F2,Y)))
     => member5262025264175285858nt_int(product_Pair_int_int(X,Y),measures_int(cons_fun_int_nat(F2,Fs))) ) ).

% measures_less
tff(fact_3538_measures__lesseq,axiom,
    ! [F2: fun_Code_integer_nat,X: code_integer,Y: code_integer,Fs: list_f8684324736568799034er_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Code_integer_nat(F2,X)),aa_Code_integer_nat(F2,Y)))
     => ( member157494554546826820nteger(produc1086072967326762835nteger(X,Y),measur8870801148506250077nteger(Fs))
       => member157494554546826820nteger(produc1086072967326762835nteger(X,Y),measur8870801148506250077nteger(cons_f3120178438656451060er_nat(F2,Fs))) ) ) ).

% measures_lesseq
tff(fact_3539_measures__lesseq,axiom,
    ! [F2: fun_Pr2602117396471556196at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Fs: list_f8256639991604547178at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(F2,X)),aa_Pro4243173770044397973at_nat(F2,Y)))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),measur2679027848233739777at_nat(Fs))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),measur2679027848233739777at_nat(cons_f462676766793977114at_nat(F2,Fs))) ) ) ).

% measures_lesseq
tff(fact_3540_measures__lesseq,axiom,
    ! [F2: fun_se4097146774670233028at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Fs: list_f2430256441796990922at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set4749074711808745205at_nat(F2,X)),aa_set4749074711808745205at_nat(F2,Y)))
     => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),measur2694323259624372065at_nat(Fs))
       => member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),measur2694323259624372065at_nat(cons_f718425424816052858at_nat(F2,Fs))) ) ) ).

% measures_lesseq
tff(fact_3541_measures__lesseq,axiom,
    ! [F2: fun_nat_nat,X: nat,Y: nat,Fs: list_fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F2,X)),aa_nat_nat(F2,Y)))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),measures_nat(Fs))
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),measures_nat(cons_fun_nat_nat(F2,Fs))) ) ) ).

% measures_lesseq
tff(fact_3542_measures__lesseq,axiom,
    ! [F2: fun_int_nat,X: int,Y: int,Fs: list_fun_int_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F2,X)),aa_int_nat(F2,Y)))
     => ( member5262025264175285858nt_int(product_Pair_int_int(X,Y),measures_int(Fs))
       => member5262025264175285858nt_int(product_Pair_int_int(X,Y),measures_int(cons_fun_int_nat(F2,Fs))) ) ) ).

% measures_lesseq
tff(fact_3543_set__replicate__Suc,axiom,
    ! [N: nat,X: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,N),X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ).

% set_replicate_Suc
tff(fact_3544_set__replicate__Suc,axiom,
    ! [N: nat,X: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,N),X)) = insert_nat(X,bot_bot_set_nat) ).

% set_replicate_Suc
tff(fact_3545_set__replicate__Suc,axiom,
    ! [N: nat,X: int] : set_int2(replicate_int(aa_nat_nat(suc,N),X)) = insert_int(X,bot_bot_set_int) ).

% set_replicate_Suc
tff(fact_3546_set__replicate__Suc,axiom,
    ! [N: nat,X: real] : set_real2(replicate_real(aa_nat_nat(suc,N),X)) = insert_real(X,bot_bot_set_real) ).

% set_replicate_Suc
tff(fact_3547_set__replicate__conv__if,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N != zero_zero_nat )
       => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% set_replicate_conv_if
tff(fact_3548_set__replicate__conv__if,axiom,
    ! [N: nat,X: nat] :
      ( ( ( N = zero_zero_nat )
       => ( set_nat2(replicate_nat(N,X)) = bot_bot_set_nat ) )
      & ( ( N != zero_zero_nat )
       => ( set_nat2(replicate_nat(N,X)) = insert_nat(X,bot_bot_set_nat) ) ) ) ).

% set_replicate_conv_if
tff(fact_3549_set__replicate__conv__if,axiom,
    ! [N: nat,X: int] :
      ( ( ( N = zero_zero_nat )
       => ( set_int2(replicate_int(N,X)) = bot_bot_set_int ) )
      & ( ( N != zero_zero_nat )
       => ( set_int2(replicate_int(N,X)) = insert_int(X,bot_bot_set_int) ) ) ) ).

% set_replicate_conv_if
tff(fact_3550_set__replicate__conv__if,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero_nat )
       => ( set_real2(replicate_real(N,X)) = bot_bot_set_real ) )
      & ( ( N != zero_zero_nat )
       => ( set_real2(replicate_real(N,X)) = insert_real(X,bot_bot_set_real) ) ) ) ).

% set_replicate_conv_if
tff(fact_3551_Cons__replicate__eq,axiom,
    ! [X: nat,Xs: list_nat,N: nat,Y: nat] :
      ( ( cons_nat(X,Xs) = replicate_nat(N,Y) )
    <=> ( ( X = Y )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( Xs = replicate_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),X) ) ) ) ).

% Cons_replicate_eq
tff(fact_3552_Cons__replicate__eq,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( cons_VEBT_VEBT(X,Xs) = replicate_VEBT_VEBT(N,Y) )
    <=> ( ( X = Y )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( Xs = replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),X) ) ) ) ).

% Cons_replicate_eq
tff(fact_3553_divmod__step__eq,axiom,
    ! [L: num,R: nat,Q2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R),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)),R))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q2)),R) ) ) ) ).

% divmod_step_eq
tff(fact_3554_divmod__step__eq,axiom,
    ! [L: num,R: int,Q2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R))
       => ( unique5024387138958732305ep_int(L,product_Pair_int_int(Q2,R)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,R),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)),R))
       => ( unique5024387138958732305ep_int(L,product_Pair_int_int(Q2,R)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2),R) ) ) ) ).

% divmod_step_eq
tff(fact_3555_divmod__step__eq,axiom,
    ! [L: num,R: code_integer,Q2: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q2,R)) = produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q2),one_one_Code_integer),minus_8373710615458151222nteger(R,numera6620942414471956472nteger(L))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q2,R)) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q2),R) ) ) ) ).

% divmod_step_eq
tff(fact_3556_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: 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)),N))
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),Ma))
          & ? [M: nat] :
              ( ( some_nat(M) = vEBT_vebt_mint(Summary) )
              & 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))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ) ) ) ).

% mintlistlength
tff(fact_3557_insert__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( 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))),N)))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vEBT_set_vebt(T)),insert_nat(X,bot_bot_set_nat)) = vEBT_set_vebt(vEBT_vebt_insert(T,X)) ) ) ) ).

% insert_correct
tff(fact_3558_insert__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( 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))),N)))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vEBT_VEBT_set_vebt(T)),insert_nat(X,bot_bot_set_nat)) = vEBT_VEBT_set_vebt(vEBT_vebt_insert(T,X)) ) ) ) ).

% insert_corr
tff(fact_3559_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => 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))),N)),one_one_nat)))) ) ).

% inrange
tff(fact_3560_nat__bit__induct,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P2,zero_zero_nat))
     => ( ! [N3: nat] :
            ( pp(aa_nat_bool(P2,N3))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
             => pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))) ) )
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(P2,N3))
             => pp(aa_nat_bool(P2,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))),N3)))) )
         => pp(aa_nat_bool(P2,N)) ) ) ) ).

% nat_bit_induct
tff(fact_3561_nat__induct2,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P2,zero_zero_nat))
     => ( pp(aa_nat_bool(P2,one_one_nat))
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(P2,N3))
             => pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
         => pp(aa_nat_bool(P2,N)) ) ) ) ).

% nat_induct2
tff(fact_3562_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) != 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,N),M2)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3563_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M2: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != 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,N),M2)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3564_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N: 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,M2),N)) != 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))),M2) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_3565_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N: 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,M2),N)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_3566_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_3567_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_3568_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_3569_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_3570_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_3571_bits__div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_3572_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_3573_div__by__0,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_3574_div__by__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_3575_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_3576_div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_3577_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_3578_div__0,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(zero_zero_real),A) = zero_zero_real ).

% div_0
tff(fact_3579_div__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,zero_zero_rat),A) = zero_zero_rat ).

% div_0
tff(fact_3580_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_3581_div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_3582_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_3583_division__ring__divide__zero,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_3584_division__ring__divide__zero,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_3585_divide__cancel__right,axiom,
    ! [A: complex,C2: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C2) = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_3586_divide__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),C2) = aa_real_real(divide_divide_real(B),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_3587_divide__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_3588_divide__cancel__left,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C2),A) = aa_complex_complex(divide1717551699836669952omplex(C2),B) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_3589_divide__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(C2),A) = aa_real_real(divide_divide_real(C2),B) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_3590_divide__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),A) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),B) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_3591_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_3592_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_3593_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_3594_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),C2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),C2) ).

% times_divide_eq_right
tff(fact_3595_times__divide__eq__right,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(divide_divide_real(B),C2)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C2) ).

% times_divide_eq_right
tff(fact_3596_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C2) ).

% times_divide_eq_right
tff(fact_3597_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(divide1717551699836669952omplex(B),C2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C2)),B) ).

% divide_divide_eq_right
tff(fact_3598_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(divide_divide_real(A),aa_real_real(divide_divide_real(B),C2)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B) ).

% divide_divide_eq_right
tff(fact_3599_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B) ).

% divide_divide_eq_right
tff(fact_3600_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C2) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(B),C2)) ).

% divide_divide_eq_left
tff(fact_3601_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(A),B)),C2) = aa_real_real(divide_divide_real(A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ).

% divide_divide_eq_left
tff(fact_3602_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ).

% divide_divide_eq_left
tff(fact_3603_times__divide__eq__left,axiom,
    ! [B: complex,C2: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(B),C2)),A) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(B),A)),C2) ).

% times_divide_eq_left
tff(fact_3604_times__divide__eq__left,axiom,
    ! [B: real,C2: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(B),C2)),A) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),C2) ).

% times_divide_eq_left
tff(fact_3605_times__divide__eq__left,axiom,
    ! [B: rat,C2: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),A) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),C2) ).

% times_divide_eq_left
tff(fact_3606_div__by__1,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,one_one_Code_integer) = A ).

% div_by_1
tff(fact_3607_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_3608_div__by__1,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),one_one_real) = A ).

% div_by_1
tff(fact_3609_div__by__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),one_one_rat) = A ).

% div_by_1
tff(fact_3610_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_3611_div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% div_by_1
tff(fact_3612_atLeastAtMost__iff,axiom,
    ! [I2: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I2,set_or4548717258645045905et_nat(L,U))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(L),I2))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(I2),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3613_atLeastAtMost__iff,axiom,
    ! [I2: rat,L: rat,U: rat] :
      ( member_rat(I2,set_or633870826150836451st_rat(L,U))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),I2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I2),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3614_atLeastAtMost__iff,axiom,
    ! [I2: num,L: num,U: num] :
      ( member_num(I2,set_or7049704709247886629st_num(L,U))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),I2))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I2),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3615_atLeastAtMost__iff,axiom,
    ! [I2: nat,L: nat,U: nat] :
      ( pp(aa_set_nat_bool(member_nat(I2),aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),I2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3616_atLeastAtMost__iff,axiom,
    ! [I2: int,L: int,U: int] :
      ( pp(aa_set_int_bool(member_int(I2),set_or1266510415728281911st_int(L,U)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),I2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3617_atLeastAtMost__iff,axiom,
    ! [I2: real,L: real,U: real] :
      ( pp(aa_set_real_bool(member_real(I2),set_or1222579329274155063t_real(L,U)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),I2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I2),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_3618_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_3619_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_3620_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_3621_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_3622_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_3623_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_3624_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_3625_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_3626_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_3627_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_3628_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_3629_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_3630_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_3631_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_3632_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_3633_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_3634_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_3635_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( ( C2 = zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(C2),B)) = zero_zero_complex ) )
      & ( ( C2 != zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(C2),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_3636_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ( C2 = zero_zero_real )
       => ( aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) = zero_zero_real ) )
      & ( ( C2 != zero_zero_real )
       => ( aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) = aa_real_real(divide_divide_real(A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_3637_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ( C2 = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) = zero_zero_rat ) )
      & ( ( C2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_3638_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(C2),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_3639_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_3640_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_3641_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(B),C2)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_3642_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_3643_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_3644_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C2)),aa_complex_complex(times_times_complex(B),C2)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_3645_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_3646_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_3647_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C2)),aa_complex_complex(times_times_complex(C2),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_3648_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_3649_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_3650_div__self,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(A,A) = one_one_Code_integer ) ) ).

% div_self
tff(fact_3651_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_3652_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ).

% div_self
tff(fact_3653_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),A) = one_one_rat ) ) ).

% div_self
tff(fact_3654_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_3655_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),A) = one_one_int ) ) ).

% div_self
tff(fact_3656_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(divide_divide_real(one_one_real),A) )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_3657_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_3658_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(divide_divide_real(one_one_real),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_3659_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_3660_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real = aa_real_real(divide_divide_real(B),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_3661_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_3662_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(divide_divide_real(B),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_3663_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_3664_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_3665_divide__self__if,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(divide_divide_real(A),A) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ) ).

% divide_self_if
tff(fact_3666_divide__self__if,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),A) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),A) = one_one_rat ) ) ) ).

% divide_self_if
tff(fact_3667_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_3668_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ).

% divide_self
tff(fact_3669_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_3670_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_3671_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real = aa_real_real(divide_divide_real(A),B) )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_3672_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_3673_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_3674_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),B) = one_one_real )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_3675_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_3676_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_3677_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_3678_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_3679_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_3680_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_3681_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_3682_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_3683_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_3684_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_3685_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_3686_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_3687_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_3688_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat,D: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(A,B),set_or4548717258645045905et_nat(C2,D))
    <=> ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
        | ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C2),A))
          & pp(aa_set_nat_bool(ord_less_eq_set_nat(B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3689_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B),set_or633870826150836451st_rat(C2,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,C2),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3690_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( ord_less_eq_set_num(set_or7049704709247886629st_num(A,B),set_or7049704709247886629st_num(C2,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,C2),A))
          & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3691_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C2: 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(C2),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,C2),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3692_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C2,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,C2),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3693_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C2,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,C2),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3694_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_3695_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_3696_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_3697_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_3698_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_3699_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_3700_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_set_real_bool(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_3701_atLeastAtMost__singleton,axiom,
    ! [A: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(A),A) = insert_nat(A,bot_bot_set_nat) ).

% atLeastAtMost_singleton
tff(fact_3702_atLeastAtMost__singleton,axiom,
    ! [A: int] : set_or1266510415728281911st_int(A,A) = insert_int(A,bot_bot_set_int) ).

% atLeastAtMost_singleton
tff(fact_3703_atLeastAtMost__singleton,axiom,
    ! [A: real] : set_or1222579329274155063t_real(A,A) = insert_real(A,bot_bot_set_real) ).

% atLeastAtMost_singleton
tff(fact_3704_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = insert_nat(C2,bot_bot_set_nat) )
    <=> ( ( A = B )
        & ( B = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_3705_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( set_or1266510415728281911st_int(A,B) = insert_int(C2,bot_bot_set_int) )
    <=> ( ( A = B )
        & ( B = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_3706_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( set_or1222579329274155063t_real(A,B) = insert_real(C2,bot_bot_set_real) )
    <=> ( ( A = B )
        & ( B = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_3707_nat__mult__div__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ( K2 = 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,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)) = zero_zero_nat ) )
      & ( ( K2 != 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,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_3708_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(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_3709_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(aa_rat_fun_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_3710_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_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_3711_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_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_3712_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_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_3713_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(aa_rat_fun_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_3714_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(divide_divide_real(B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% less_divide_eq_1_pos
tff(fact_3715_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(aa_rat_fun_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_3716_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(divide_divide_real(B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% less_divide_eq_1_neg
tff(fact_3717_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(aa_rat_fun_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_3718_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(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_3719_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(aa_rat_fun_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_3720_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(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_3721_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(aa_rat_fun_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_3722_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_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_3723_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_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_3724_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(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_3725_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(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_3726_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(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_3727_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(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2))),B)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_3728_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_3729_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( aa_real_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_3730_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_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(aa_rat_fun_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_3731_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_3732_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A = aa_real_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_3733_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A = aa_rat_rat(aa_rat_fun_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(aa_rat_fun_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_3734_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(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_3735_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(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2))),B)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_3736_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(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_3737_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(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_3738_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_3739_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(divide_divide_real(one_one_real),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_3740_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_3741_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_3742_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(divide_divide_real(one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_3743_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_3744_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),aa_nat_real(semiri5074537144036343181t_real,M2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),M2)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_3745_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_3746_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(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_3747_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(aa_rat_fun_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_3748_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(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_3749_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(aa_rat_fun_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_3750_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(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_3751_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(aa_rat_fun_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_3752_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(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_3753_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(aa_rat_fun_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_3754_bits__1__div__2,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% bits_1_div_2
tff(fact_3755_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_3756_bits__1__div__2,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_3757_decr__mult__lemma,axiom,
    ! [D: int,P2: fun_int_bool,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X3: int] :
            ( pp(aa_int_bool(P2,X3))
           => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2))
         => ! [X5: int] :
              ( pp(aa_int_bool(P2,X5))
             => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_3758_minusinfinity,axiom,
    ! [D: int,P1: fun_int_bool,P2: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X3: int,K: int] :
            ( pp(aa_int_bool(P1,X3))
          <=> pp(aa_int_bool(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) )
       => ( ? [Z4: int] :
            ! [X3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z4))
             => ( pp(aa_int_bool(P2,X3))
              <=> pp(aa_int_bool(P1,X3)) ) )
         => ( ? [X_1: int] : pp(aa_int_bool(P1,X_1))
           => ? [X_12: int] : pp(aa_int_bool(P2,X_12)) ) ) ) ) ).

% minusinfinity
tff(fact_3759_plusinfinity,axiom,
    ! [D: int,P3: fun_int_bool,P2: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X3: int,K: int] :
            ( pp(aa_int_bool(P3,X3))
          <=> pp(aa_int_bool(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) )
       => ( ? [Z4: int] :
            ! [X3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X3))
             => ( pp(aa_int_bool(P2,X3))
              <=> pp(aa_int_bool(P3,X3)) ) )
         => ( ? [X_1: int] : pp(aa_int_bool(P3,X_1))
           => ? [X_12: int] : pp(aa_int_bool(P2,X_12)) ) ) ) ) ).

% plusinfinity
tff(fact_3760_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_3761_times__int__code_I1_J,axiom,
    ! [K2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,K2),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_3762_pos__zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M2))
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M2),N) = one_one_int )
      <=> ( ( M2 = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_3763_zmult__zless__mono2,axiom,
    ! [I2: int,J2: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),J2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I2)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),J2))) ) ) ).

% zmult_zless_mono2
tff(fact_3764_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q4: int,R4: int,Q2: int,R: 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),Q4)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R))
         => ( 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,Q2),Q4)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3765_unique__quotient__lemma,axiom,
    ! [B: int,Q4: int,R4: int,Q2: int,R: 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),Q4)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R)))
     => ( 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,R),B))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q4),Q2)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3766_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R: int,B2: int,Q4: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q4)),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,B2),Q4)),R4)),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),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),B2))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q4),Q2)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3767_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R: int,B2: int,Q4: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q4)),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,B2),Q4)),R4)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q2),Q4)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3768_q__pos__lemma,axiom,
    ! [B2: int,Q4: 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,B2),Q4)),R4)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Q4)) ) ) ) ).

% q_pos_lemma
tff(fact_3769_incr__mult__lemma,axiom,
    ! [D: int,P2: fun_int_bool,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X3: int] :
            ( pp(aa_int_bool(P2,X3))
           => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2))
         => ! [X5: int] :
              ( pp(aa_int_bool(P2,X5))
             => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3770_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: rat,M2: rat,U: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),M2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,M2),U))
       => ( sup_sup_set_rat(set_or633870826150836451st_rat(L,M2),set_or633870826150836451st_rat(M2,U)) = set_or633870826150836451st_rat(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_3771_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: num,M2: num,U: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),M2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),U))
       => ( sup_sup_set_num(set_or7049704709247886629st_num(L,M2),set_or7049704709247886629st_num(M2,U)) = set_or7049704709247886629st_num(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_3772_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: nat,M2: nat,U: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),U))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),M2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),U)) = aa_nat_set_nat(set_or1269000886237332187st_nat(L),U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_3773_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: int,M2: int,U: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),M2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M2),U))
       => ( sup_sup_set_int(set_or1266510415728281911st_int(L,M2),set_or1266510415728281911st_int(M2,U)) = set_or1266510415728281911st_int(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_3774_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: real,M2: real,U: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),M2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M2),U))
       => ( sup_sup_set_real(set_or1222579329274155063t_real(L,M2),set_or1222579329274155063t_real(M2,U)) = set_or1222579329274155063t_real(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_3775_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_3776_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_3777_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_3778_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_3779_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C2: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C2) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(C2),B)) ).

% divide_divide_eq_left'
tff(fact_3780_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(A),B)),C2) = aa_real_real(divide_divide_real(A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) ).

% divide_divide_eq_left'
tff(fact_3781_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) ).

% divide_divide_eq_left'
tff(fact_3782_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_3783_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z2: real,W2: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(X),Y)),aa_real_real(divide_divide_real(Z2),W2)) = aa_real_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_3784_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z2: rat,W2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Z2),W2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),W2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z2)) ).

% divide_divide_times_eq
tff(fact_3785_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_3786_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(divide_divide_real(X),Y)),aa_real_real(divide_divide_real(Z2),W2)) = aa_real_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_3787_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z2: rat,W2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Z2),W2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),W2)) ).

% times_divide_times_eq
tff(fact_3788_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat,D: set_nat] :
      ( ord_less_set_set_nat(set_or4548717258645045905et_nat(A,B),set_or4548717258645045905et_nat(C2,D))
    <=> ( ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
          | ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C2),A))
            & pp(aa_set_nat_bool(ord_less_eq_set_nat(B),D))
            & ( pp(aa_set_nat_bool(ord_less_set_nat(C2),A))
              | pp(aa_set_nat_bool(ord_less_set_nat(B),D)) ) ) )
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(C2),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3789_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ord_less_set_rat(set_or633870826150836451st_rat(A,B),set_or633870826150836451st_rat(C2,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,C2),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,C2),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,C2),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3790_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( ord_less_set_num(set_or7049704709247886629st_num(A,B),set_or7049704709247886629st_num(C2,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,C2),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,C2),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,C2),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3791_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C2: 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(C2),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,C2),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,C2),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,C2),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3792_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C2,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,C2),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,C2),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,C2),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3793_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C2,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,C2),A))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D))
            & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),A))
              | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),D)) ) ) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3794_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_3795_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_set_real_bool(finite_finite_real,set_or1222579329274155063t_real(A,B))) ) ).

% infinite_Icc
tff(fact_3796_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = insert_nat(A,bot_bot_set_nat) ) ) ).

% atLeastAtMost_singleton'
tff(fact_3797_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( set_or1266510415728281911st_int(A,B) = insert_int(A,bot_bot_set_int) ) ) ).

% atLeastAtMost_singleton'
tff(fact_3798_atLeastAtMost__singleton_H,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( set_or1222579329274155063t_real(A,B) = insert_real(A,bot_bot_set_real) ) ) ).

% atLeastAtMost_singleton'
tff(fact_3799_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C2: 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,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B),C2)),aa_real_real(divide_divide_real(A),C2))) ) ) ).

% divide_right_mono_neg
tff(fact_3800_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C2: 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,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2))) ) ) ).

% divide_right_mono_neg
tff(fact_3801_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(divide_divide_real(X),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_3802_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_3803_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(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_nonneg
tff(fact_3804_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_nonneg
tff(fact_3805_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(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_nonpos
tff(fact_3806_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_nonpos
tff(fact_3807_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(divide_divide_real(X),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_3808_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_3809_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(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_3810_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(aa_rat_fun_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_3811_divide__right__mono,axiom,
    ! [A: real,B: real,C2: 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),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(A),C2)),aa_real_real(divide_divide_real(B),C2))) ) ) ).

% divide_right_mono
tff(fact_3812_divide__right__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))) ) ) ).

% divide_right_mono
tff(fact_3813_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_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_3814_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(aa_rat_fun_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_3815_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(A),C2)),aa_real_real(divide_divide_real(B),C2))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_3816_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: 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,C2),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_3817_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C2: 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),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(A),C2)),aa_real_real(divide_divide_real(B),C2))) ) ) ).

% divide_strict_right_mono
tff(fact_3818_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))) ) ) ).

% divide_strict_right_mono
tff(fact_3819_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(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_3820_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(aa_rat_fun_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_3821_divide__less__cancel,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(A),C2)),aa_real_real(divide_divide_real(B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) )
        & ( C2 != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_3822_divide__less__cancel,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => 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,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
        & ( C2 != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_3823_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_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_3824_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_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_3825_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(divide_divide_real(X),Y))) ) ) ).

% divide_pos_pos
tff(fact_3826_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Y))) ) ) ).

% divide_pos_pos
tff(fact_3827_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(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_pos_neg
tff(fact_3828_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat)) ) ) ).

% divide_pos_neg
tff(fact_3829_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(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_neg_pos
tff(fact_3830_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat)) ) ) ).

% divide_neg_pos
tff(fact_3831_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(divide_divide_real(X),Y))) ) ) ).

% divide_neg_neg
tff(fact_3832_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Y))) ) ) ).

% divide_neg_neg
tff(fact_3833_all__nat__less,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( ! [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N))
         => pp(aa_nat_bool(P2,M6)) )
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
         => pp(aa_nat_bool(P2,X4)) ) ) ).

% all_nat_less
tff(fact_3834_ex__nat__less,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( ? [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N))
          & pp(aa_nat_bool(P2,M6)) )
    <=> ? [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
          & pp(aa_nat_bool(P2,X4)) ) ) ).

% ex_nat_less
tff(fact_3835_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_3836_frac__eq__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( aa_real_real(divide_divide_real(X),Y) = aa_real_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_3837_frac__eq__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z2) )
        <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_3838_divide__eq__eq,axiom,
    ! [B: complex,C2: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C2) = A )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),C2) ) )
        & ( ( C2 = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
tff(fact_3839_divide__eq__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( aa_real_real(divide_divide_real(B),C2) = A )
    <=> ( ( ( C2 != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) ) )
        & ( ( C2 = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
tff(fact_3840_divide__eq__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2) = A )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) ) )
        & ( ( C2 = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
tff(fact_3841_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C2: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),C2) = B ) )
        & ( ( C2 = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
tff(fact_3842_eq__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( A = aa_real_real(divide_divide_real(B),C2) )
    <=> ( ( ( C2 != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = B ) )
        & ( ( C2 = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
tff(fact_3843_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2) )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = B ) )
        & ( ( C2 = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
tff(fact_3844_divide__eq__imp,axiom,
    ! [C2: complex,B: complex,A: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( B = aa_complex_complex(times_times_complex(A),C2) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B),C2) = A ) ) ) ).

% divide_eq_imp
tff(fact_3845_divide__eq__imp,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( C2 != zero_zero_real )
     => ( ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) )
       => ( aa_real_real(divide_divide_real(B),C2) = A ) ) ) ).

% divide_eq_imp
tff(fact_3846_divide__eq__imp,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2) = A ) ) ) ).

% divide_eq_imp
tff(fact_3847_eq__divide__imp,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C2) = B )
       => ( A = aa_complex_complex(divide1717551699836669952omplex(B),C2) ) ) ) ).

% eq_divide_imp
tff(fact_3848_eq__divide__imp,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = B )
       => ( A = aa_real_real(divide_divide_real(B),C2) ) ) ) ).

% eq_divide_imp
tff(fact_3849_eq__divide__imp,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = B )
       => ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2) ) ) ) ).

% eq_divide_imp
tff(fact_3850_nonzero__divide__eq__eq,axiom,
    ! [C2: complex,B: complex,A: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B),C2) = A )
      <=> ( B = aa_complex_complex(times_times_complex(A),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3851_nonzero__divide__eq__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(divide_divide_real(B),C2) = A )
      <=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3852_nonzero__divide__eq__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2) = A )
      <=> ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3853_nonzero__eq__divide__eq,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
      <=> ( aa_complex_complex(times_times_complex(A),C2) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3854_nonzero__eq__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( A = aa_real_real(divide_divide_real(B),C2) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3855_nonzero__eq__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3856_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_3857_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(divide_divide_real(A),B) = one_one_real )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_3858_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_3859_divide__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_3860_divide__numeral__1,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),aa_num_real(numeral_numeral_real,one)) = A ).

% divide_numeral_1
tff(fact_3861_divide__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% divide_numeral_1
tff(fact_3862_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_3863_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(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ).

% div_positive
tff(fact_3864_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_3865_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(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3866_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(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_pos
tff(fact_3867_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_pos
tff(fact_3868_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(divide_divide_real(X),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_3869_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_3870_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(divide_divide_real(X),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_3871_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_3872_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(divide_divide_real(X),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_neg
tff(fact_3873_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_neg
tff(fact_3874_divide__le__cancel,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(A),C2)),aa_real_real(divide_divide_real(B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% divide_le_cancel
tff(fact_3875_divide__le__cancel,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => 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,C2),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% divide_le_cancel
tff(fact_3876_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(divide_divide_real(X),Z2)),aa_real_real(divide_divide_real(Y),W2))) ) ) ) ) ).

% frac_less2
tff(fact_3877_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Z2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W2))) ) ) ) ) ).

% frac_less2
tff(fact_3878_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(divide_divide_real(X),Z2)),aa_real_real(divide_divide_real(Y),W2))) ) ) ) ) ).

% frac_less
tff(fact_3879_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Z2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W2))) ) ) ) ) ).

% frac_less
tff(fact_3880_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(divide_divide_real(X),Z2)),aa_real_real(divide_divide_real(Y),W2))) ) ) ) ) ).

% frac_le
tff(fact_3881_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Z2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W2))) ) ) ) ) ).

% frac_le
tff(fact_3882_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3883_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3884_divide__less__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B),C2)),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_3885_divide__less__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_3886_less__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(divide_divide_real(B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_3887_less__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_3888_neg__divide__less__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B),C2)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B)) ) ) ).

% neg_divide_less_eq
tff(fact_3889_neg__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B)) ) ) ).

% neg_divide_less_eq
tff(fact_3890_neg__less__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(divide_divide_real(B),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) ) ) ).

% neg_less_divide_eq
tff(fact_3891_neg__less__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2))) ) ) ).

% neg_less_divide_eq
tff(fact_3892_pos__divide__less__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B),C2)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) ) ) ).

% pos_divide_less_eq
tff(fact_3893_pos__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2))) ) ) ).

% pos_divide_less_eq
tff(fact_3894_pos__less__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(divide_divide_real(B),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B)) ) ) ).

% pos_less_divide_eq
tff(fact_3895_pos__less__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B)) ) ) ).

% pos_less_divide_eq
tff(fact_3896_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(divide_divide_real(X),Y)),Z2)) ) ) ).

% mult_imp_div_pos_less
tff(fact_3897_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(aa_rat_fun_rat_rat(times_times_rat,Z2),Y)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),Z2)) ) ) ).

% mult_imp_div_pos_less
tff(fact_3898_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(divide_divide_real(X),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_3899_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(aa_rat_fun_rat_rat(times_times_rat,Z2),Y)),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_3900_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C2: 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),C2))
       => ( 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(divide_divide_real(C2),A)),aa_real_real(divide_divide_real(C2),B))) ) ) ) ).

% divide_strict_left_mono
tff(fact_3901_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C2: 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),C2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),B))) ) ) ) ).

% divide_strict_left_mono
tff(fact_3902_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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(divide_divide_real(C2),A)),aa_real_real(divide_divide_real(C2),B))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3903_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C2: 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,C2),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),B))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3904_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(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_3905_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(aa_rat_fun_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_3906_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_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_3907_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_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_3908_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C2: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C2) = numera6690914467698888265omplex(W2) )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C2) ) )
        & ( ( C2 = zero_zero_complex )
         => ( numera6690914467698888265omplex(W2) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3909_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( ( aa_real_real(divide_divide_real(B),C2) = aa_num_real(numeral_numeral_real,W2) )
    <=> ( ( ( C2 != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2) ) )
        & ( ( C2 = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3910_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2) = aa_num_rat(numeral_numeral_rat,W2) )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2) ) )
        & ( ( C2 = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3911_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: complex,C2: complex] :
      ( ( numera6690914467698888265omplex(W2) = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C2) = B ) )
        & ( ( C2 = zero_zero_complex )
         => ( numera6690914467698888265omplex(W2) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3912_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( ( aa_num_real(numeral_numeral_real,W2) = aa_real_real(divide_divide_real(B),C2) )
    <=> ( ( ( C2 != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C2) = B ) )
        & ( ( C2 = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3913_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2) )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2) = B ) )
        & ( ( C2 = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3914_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_3915_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(A),Z2)),B) = B ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(A),Z2)),B) = aa_real_real(divide_divide_real(aa_real_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_3916_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z2)),B) = B ) )
      & ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z2)),B) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_3917_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_3918_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(plus_plus_real(A),aa_real_real(divide_divide_real(B),Z2)) = A ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(plus_plus_real(A),aa_real_real(divide_divide_real(B),Z2)) = aa_real_real(divide_divide_real(aa_real_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_3919_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Z2)) = A ) )
      & ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Z2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z2)),B)),Z2) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_3920_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_3921_add__frac__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(X),Y)),aa_real_real(divide_divide_real(W2),Z2)) = aa_real_real(divide_divide_real(aa_real_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_3922_add__frac__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z2)) ) ) ) ).

% add_frac_eq
tff(fact_3923_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_3924_add__frac__num,axiom,
    ! [Y: real,X: real,Z2: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(X),Y)),Z2) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),Y))),Y) ) ) ).

% add_frac_num
tff(fact_3925_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z2: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),Z2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z2),Y))),Y) ) ) ).

% add_frac_num
tff(fact_3926_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_3927_add__num__frac,axiom,
    ! [Y: real,Z2: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(plus_plus_real(Z2),aa_real_real(divide_divide_real(X),Y)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),Y))),Y) ) ) ).

% add_num_frac
tff(fact_3928_add__num__frac,axiom,
    ! [Y: rat,Z2: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z2),Y))),Y) ) ) ).

% add_num_frac
tff(fact_3929_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_3930_add__divide__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(Y),Z2)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z2)),Y)),Z2) ) ) ).

% add_divide_eq_iff
tff(fact_3931_add__divide__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),Z2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z2)),Y)),Z2) ) ) ).

% add_divide_eq_iff
tff(fact_3932_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_3933_divide__add__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(X),Z2)),Y) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z2))),Z2) ) ) ).

% divide_add_eq_iff
tff(fact_3934_divide__add__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z2)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z2))),Z2) ) ) ).

% divide_add_eq_iff
tff(fact_3935_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_3936_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(minus_minus_real(A),aa_real_real(divide_divide_real(B),Z2)) = A ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(minus_minus_real(A),aa_real_real(divide_divide_real(B),Z2)) = aa_real_real(divide_divide_real(aa_real_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_3937_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Z2)) = A ) )
      & ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Z2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z2)),B)),Z2) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_3938_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_3939_diff__frac__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(X),Y)),aa_real_real(divide_divide_real(W2),Z2)) = aa_real_real(divide_divide_real(aa_real_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_3940_diff__frac__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z2)) ) ) ) ).

% diff_frac_eq
tff(fact_3941_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_3942_diff__divide__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(minus_minus_real(X),aa_real_real(divide_divide_real(Y),Z2)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z2)),Y)),Z2) ) ) ).

% diff_divide_eq_iff
tff(fact_3943_diff__divide__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),Z2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z2)),Y)),Z2) ) ) ).

% diff_divide_eq_iff
tff(fact_3944_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_3945_divide__diff__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(X),Z2)),Y) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z2))),Z2) ) ) ).

% divide_diff_eq_iff
tff(fact_3946_divide__diff__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z2)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z2))),Z2) ) ) ).

% divide_diff_eq_iff
tff(fact_3947_card__Un__le,axiom,
    ! [A4: set_complex,B5: set_complex] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(sup_sup_set_complex(A4,B5))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_complex(A4)),finite_card_complex(B5)))) ).

% card_Un_le
tff(fact_3948_card__Un__le,axiom,
    ! [A4: set_int,B5: set_int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(sup_sup_set_int(A4,B5))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_int(A4)),finite_card_int(B5)))) ).

% card_Un_le
tff(fact_3949_card__Un__le,axiom,
    ! [A4: set_list_nat,B5: set_list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_list_nat(sup_sup_set_list_nat(A4,B5))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(A4)),finite_card_list_nat(B5)))) ).

% card_Un_le
tff(fact_3950_card__Un__le,axiom,
    ! [A4: set_set_nat,B5: set_set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_set_nat(sup_sup_set_set_nat(A4,B5))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_set_nat(A4)),finite_card_set_nat(B5)))) ).

% card_Un_le
tff(fact_3951_card__Un__le,axiom,
    ! [A4: set_nat,B5: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B5)))) ).

% card_Un_le
tff(fact_3952_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% atLeast0_atMost_Suc
tff(fact_3953_nat__mult__div__cancel1,axiom,
    ! [K2: nat,M2: nat,N: 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(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_3954_Icc__eq__insert__lb__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N) = insert_nat(M2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_3955_atLeastAtMostSuc__conv,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_3956_atLeastAtMost__insertL,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( insert_nat(M2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N) ) ) ).

% atLeastAtMost_insertL
tff(fact_3957_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N5),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N5)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_3958_divide__le__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B),C2)),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_3959_divide__le__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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_3960_le__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_3961_le__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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_3962_divide__left__mono,axiom,
    ! [B: real,A: real,C2: 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),C2))
       => ( 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(divide_divide_real(C2),A)),aa_real_real(divide_divide_real(C2),B))) ) ) ) ).

% divide_left_mono
tff(fact_3963_divide__left__mono,axiom,
    ! [B: rat,A: rat,C2: 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),C2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),B))) ) ) ) ).

% divide_left_mono
tff(fact_3964_neg__divide__le__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B),C2)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B)) ) ) ).

% neg_divide_le_eq
tff(fact_3965_neg__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B)) ) ) ).

% neg_divide_le_eq
tff(fact_3966_neg__le__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) ) ) ).

% neg_le_divide_eq
tff(fact_3967_neg__le__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2))) ) ) ).

% neg_le_divide_eq
tff(fact_3968_pos__divide__le__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B),C2)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) ) ) ).

% pos_divide_le_eq
tff(fact_3969_pos__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2))) ) ) ).

% pos_divide_le_eq
tff(fact_3970_pos__le__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),B)) ) ) ).

% pos_le_divide_eq
tff(fact_3971_pos__le__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B)) ) ) ).

% pos_le_divide_eq
tff(fact_3972_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(divide_divide_real(X),Y)),Z2)) ) ) ).

% mult_imp_div_pos_le
tff(fact_3973_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(aa_rat_fun_rat_rat(times_times_rat,Z2),Y)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),Z2)) ) ) ).

% mult_imp_div_pos_le
tff(fact_3974_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(divide_divide_real(X),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_3975_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(aa_rat_fun_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_3976_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C2: 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,C2),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(divide_divide_real(C2),A)),aa_real_real(divide_divide_real(C2),B))) ) ) ) ).

% divide_left_mono_neg
tff(fact_3977_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C2: 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,C2),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),B))) ) ) ) ).

% divide_left_mono_neg
tff(fact_3978_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_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_3979_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(aa_rat_fun_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_3980_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(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_3981_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(aa_rat_fun_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_3982_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B),C2)),aa_num_real(numeral_numeral_real,W2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => 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)),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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)),C2)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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_3983_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),aa_num_rat(numeral_numeral_rat,W2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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_3984_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(divide_divide_real(B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => 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)),C2)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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)),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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_3985_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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_3986_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(divide_divide_real(X),Y)),aa_real_real(divide_divide_real(W2),Z2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_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_3987_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z2)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z2))),zero_zero_rat)) ) ) ) ).

% frac_le_eq
tff(fact_3988_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(divide_divide_real(X),Y)),aa_real_real(divide_divide_real(W2),Z2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(aa_real_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_3989_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(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z2)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z2))),zero_zero_rat)) ) ) ) ).

% frac_less_eq
tff(fact_3990_power__diff,axiom,
    ! [A: complex,N: nat,M2: nat] :
      ( ( A != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),M2)),aa_nat_complex(power_power_complex(A),N)) ) ) ) ).

% power_diff
tff(fact_3991_power__diff,axiom,
    ! [A: real,N: nat,M2: nat] :
      ( ( A != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ) ) ) ).

% power_diff
tff(fact_3992_power__diff,axiom,
    ! [A: rat,N: nat,M2: nat] :
      ( ( A != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(power_power_rat(A),M2)),aa_nat_rat(power_power_rat(A),N)) ) ) ) ).

% power_diff
tff(fact_3993_power__diff,axiom,
    ! [A: nat,N: nat,M2: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).

% power_diff
tff(fact_3994_power__diff,axiom,
    ! [A: int,N: nat,M2: nat] :
      ( ( A != zero_zero_int )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N)) ) ) ) ).

% power_diff
tff(fact_3995_div__geq,axiom,
    ! [N: nat,M2: 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(ord_less_nat,M2),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = 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,M2),N)),N)) ) ) ) ).

% div_geq
tff(fact_3996_four__x__squared,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% four_x_squared
tff(fact_3997_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C2: real,B: real,D: 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))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% L2_set_mult_ineq_lemma
tff(fact_3998_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B),C2)),aa_num_real(numeral_numeral_real,W2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => 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)),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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)),C2)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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_3999_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),aa_num_rat(numeral_numeral_rat,W2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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_4000_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(divide_divide_real(B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => 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)),C2)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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)),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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_4001_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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_4002_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(divide_divide_real(A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_4003_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(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_4004_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_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_4005_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(aa_rat_fun_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_4006_scaling__mono,axiom,
    ! [U: real,V2: real,R: real,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),V2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R),S2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(U),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(minus_minus_real(V2),U))),S2))),V2)) ) ) ) ).

% scaling_mono
tff(fact_4007_scaling__mono,axiom,
    ! [U: rat,V2: rat,R: rat,S2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),V2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),R))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,R),S2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,V2),U))),S2))),V2)) ) ) ) ).

% scaling_mono
tff(fact_4008_nat__approx__posE,axiom,
    ! [E: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
     => ~ ! [N3: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N3)))),E)) ) ).

% nat_approx_posE
tff(fact_4009_nat__approx__posE,axiom,
    ! [E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
     => ~ ! [N3: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N3)))),E)) ) ).

% nat_approx_posE
tff(fact_4010_inverse__of__nat__le,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( ( N != zero_zero_nat )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,M2))),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_4011_inverse__of__nat__le,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( ( N != zero_zero_nat )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M2))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_4012_triangle__def,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_4013_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,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(divide_divide_real(aa_real_real(plus_plus_real(X),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_4014_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(aa_rat_fun_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(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_4015_double__not__eq__Suc__double,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) != 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)) ).

% double_not_eq_Suc_double
tff(fact_4016_Suc__double__not__eq__double,axiom,
    ! [M2: nat,N: 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))),M2)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% Suc_double_not_eq_double
tff(fact_4017_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N: 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,M2),N)) != 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))),N) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_4018_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N: 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,M2),N)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_4019_one__div__two__eq__zero,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% one_div_two_eq_zero
tff(fact_4020_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_4021_one__div__two__eq__zero,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_4022_div2__Suc__Suc,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,M2))),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,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_4023_insert_H__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( vEBT_set_vebt(vEBT_VEBT_insert(T,X)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vEBT_set_vebt(T)),insert_nat(X,bot_bot_set_nat))),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))),N)),one_one_nat))) ) ) ).

% insert'_correct
tff(fact_4024_div__mult__self__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)),N) = M2 ) ) ).

% div_mult_self_is_m
tff(fact_4025_div__mult__self1__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M2)),N) = M2 ) ) ).

% div_mult_self1_is_m
tff(fact_4026_div__mult__self1,axiom,
    ! [B: nat,A: nat,C2: 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,C2),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_4027_div__mult__self1,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self1
tff(fact_4028_div__mult__self2,axiom,
    ! [B: nat,A: nat,C2: 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),C2))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_4029_div__mult__self2,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self2
tff(fact_4030_div__mult__self3,axiom,
    ! [B: nat,C2: 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,C2),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_4031_div__mult__self3,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self3
tff(fact_4032_zdiv__numeral__Bit0,axiom,
    ! [V2: num,W2: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V2)),aa_num_int(numeral_numeral_int,W2)) ).

% zdiv_numeral_Bit0
tff(fact_4033_real__divide__square__eq,axiom,
    ! [R: real,A: real] : aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),R)) = aa_real_real(divide_divide_real(A),R) ).

% real_divide_square_eq
tff(fact_4034_div__mult__mult1__if,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ( C2 = 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) = zero_zero_nat ) )
      & ( ( C2 != 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_mult_mult1_if
tff(fact_4035_div__mult__mult1__if,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ( C2 = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) = zero_zero_int ) )
      & ( ( C2 != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% div_mult_mult1_if
tff(fact_4036_div__mult__mult2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_4037_div__mult__mult2,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult2
tff(fact_4038_div__mult__mult1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_4039_div__mult__mult1,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult1
tff(fact_4040_div__by__Suc__0,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_nat_nat(suc,zero_zero_nat)) = M2 ).

% div_by_Suc_0
tff(fact_4041_div__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = zero_zero_nat ) ) ).

% div_less
tff(fact_4042_div__mult__self4,axiom,
    ! [B: nat,C2: 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),C2)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_4043_div__mult__self4,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self4
tff(fact_4044_zdiv__zmult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_4045_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(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)) ) ) ).

% div_neg_pos_less0
tff(fact_4046_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(aa_int_fun_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_4047_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(aa_int_fun_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_4048_zdiv__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zdiv_int
tff(fact_4049_atLeastAtMostPlus1__int__conv,axiom,
    ! [M2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M2),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)))
     => ( set_or1266510415728281911st_int(M2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N),set_or1266510415728281911st_int(M2,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_4050_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(aa_int_fun_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_4051_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(aa_int_fun_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_4052_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(aa_int_fun_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_4053_pos__imp__zdiv__pos__iff,axiom,
    ! [K2: int,I2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I2),K2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),I2)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_4054_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(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_4055_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(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_4056_div__positive__int,axiom,
    ! [L: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K2))
     => ( 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(aa_int_fun_int_int(divide_divide_int,K2),L))) ) ) ).

% div_positive_int
tff(fact_4057_div__int__pos__iff,axiom,
    ! [K2: int,L: 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(divide_divide_int,K2),L)))
    <=> ( ( K2 = zero_zero_int )
        | ( L = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2))
          & 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,K2),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_4058_zdiv__mono2__neg,axiom,
    ! [A: int,B2: 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),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_4059_zdiv__mono1__neg,axiom,
    ! [A: int,A2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A2))
     => ( 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(aa_int_fun_int_int(divide_divide_int,A2),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ).

% zdiv_mono1_neg
tff(fact_4060_zdiv__eq__0__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I2),K2) = zero_zero_int )
    <=> ( ( K2 = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),K2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),I2)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_4061_zdiv__mono2,axiom,
    ! [A: int,B2: 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),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2))) ) ) ) ).

% zdiv_mono2
tff(fact_4062_zdiv__mono1,axiom,
    ! [A: int,A2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A2))
     => ( 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(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B))) ) ) ).

% zdiv_mono1
tff(fact_4063_int__div__less__self,axiom,
    ! [X: int,K2: 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),K2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),K2)),X)) ) ) ).

% int_div_less_self
tff(fact_4064_periodic__finite__ex,axiom,
    ! [D: int,P2: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X3: int,K: int] :
            ( pp(aa_int_bool(P2,X3))
          <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) )
       => ( ? [X_13: int] : pp(aa_int_bool(P2,X_13))
        <=> ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),set_or1266510415728281911st_int(one_one_int,D)))
              & pp(aa_int_bool(P2,X4)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_4065_split__zdiv,axiom,
    ! [P2: fun_int_bool,N: int,K2: int] :
      ( pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K2)))
    <=> ( ( ( K2 = zero_zero_int )
         => pp(aa_int_bool(P2,zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
         => ! [I: int,J: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),K2))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J) ) )
             => pp(aa_int_bool(P2,I)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
         => ! [I: int,J: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),J))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),zero_zero_int))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J) ) )
             => pp(aa_int_bool(P2,I)) ) ) ) ) ).

% split_zdiv
tff(fact_4066_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_4067_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),B))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_4068_cpmi,axiom,
    ! [D3: int,P2: fun_int_bool,P3: fun_int_bool,B5: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z4))
           => ( pp(aa_int_bool(P2,X3))
            <=> pp(aa_int_bool(P3,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
                 => ! [Xb3: int] :
                      ( pp(aa_set_int_bool(member_int(Xb3),B5))
                     => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa) ) ) )
             => ( pp(aa_int_bool(P2,X3))
               => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3))) ) )
         => ( ! [X3: int,K: int] :
                ( pp(aa_int_bool(P3,X3))
              <=> pp(aa_int_bool(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3)))) )
           => ( ? [X_13: int] : pp(aa_int_bool(P2,X_13))
            <=> ( ? [X4: int] :
                    ( pp(aa_set_int_bool(member_int(X4),set_or1266510415728281911st_int(one_one_int,D3)))
                    & pp(aa_int_bool(P3,X4)) )
                | ? [X4: int] :
                    ( pp(aa_set_int_bool(member_int(X4),set_or1266510415728281911st_int(one_one_int,D3)))
                    & ? [Xa3: int] :
                        ( pp(aa_set_int_bool(member_int(Xa3),B5))
                        & pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X4))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_4069_cppi,axiom,
    ! [D3: int,P2: fun_int_bool,P3: fun_int_bool,A4: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X3))
           => ( pp(aa_int_bool(P2,X3))
            <=> pp(aa_int_bool(P3,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
                 => ! [Xb3: int] :
                      ( pp(aa_set_int_bool(member_int(Xb3),A4))
                     => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa) ) ) )
             => ( pp(aa_int_bool(P2,X3))
               => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3))) ) )
         => ( ! [X3: int,K: int] :
                ( pp(aa_int_bool(P3,X3))
              <=> pp(aa_int_bool(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3)))) )
           => ( ? [X_13: int] : pp(aa_int_bool(P2,X_13))
            <=> ( ? [X4: int] :
                    ( pp(aa_set_int_bool(member_int(X4),set_or1266510415728281911st_int(one_one_int,D3)))
                    & pp(aa_int_bool(P3,X4)) )
                | ? [X4: int] :
                    ( pp(aa_set_int_bool(member_int(X4),set_or1266510415728281911st_int(one_one_int,D3)))
                    & ? [Xa3: int] :
                        ( pp(aa_set_int_bool(member_int(Xa3),A4))
                        & pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X4))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_4070_int__power__div__base,axiom,
    ! [M2: nat,K2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(K2),M2)),K2) = aa_nat_int(power_power_int(K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_4071_div__pos__geq,axiom,
    ! [L: int,K2: 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),K2))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_4072_div__le__dividend,axiom,
    ! [M2: nat,N: 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,M2),N)),M2)) ).

% div_le_dividend
tff(fact_4073_div__le__mono,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),K2))) ) ).

% div_le_mono
tff(fact_4074_div__mult2__eq,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),Q2) ).

% div_mult2_eq
tff(fact_4075_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(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_4076_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(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_4077_div__mult2__eq_H,axiom,
    ! [A: int,M2: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% div_mult2_eq'
tff(fact_4078_div__mult2__eq_H,axiom,
    ! [A: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M2))),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% div_mult2_eq'
tff(fact_4079_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_4080_Suc__div__le__mono,axiom,
    ! [M2: nat,N: 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,M2),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M2)),N))) ).

% Suc_div_le_mono
tff(fact_4081_less__mult__imp__div__less,axiom,
    ! [M2: nat,I2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),I2)) ) ).

% less_mult_imp_div_less
tff(fact_4082_times__div__less__eq__dividend,axiom,
    ! [N: nat,M2: 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,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N))),M2)) ).

% times_div_less_eq_dividend
tff(fact_4083_div__times__less__eq__dividend,axiom,
    ! [M2: nat,N: 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,M2),N)),N)),M2)) ).

% div_times_less_eq_dividend
tff(fact_4084_div__mult2__numeral__eq,axiom,
    ! [A: nat,K2: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,K2))),aa_num_nat(numeral_numeral_nat,L)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,K2),L))) ).

% div_mult2_numeral_eq
tff(fact_4085_div__mult2__numeral__eq,axiom,
    ! [A: int,K2: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,K2),L))) ).

% div_mult2_numeral_eq
tff(fact_4086_div__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(B,A),B) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,B),one_one_Code_integer) ) ) ).

% div_add_self1
tff(fact_4087_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_4088_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_4089_div__add__self2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),B) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,B),one_one_Code_integer) ) ) ).

% div_add_self2
tff(fact_4090_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_4091_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_4092_div__le__mono2,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K2),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K2),M2))) ) ) ).

% div_le_mono2
tff(fact_4093_div__greater__zero__iff,axiom,
    ! [M2: nat,N: 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,M2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% div_greater_zero_iff
tff(fact_4094_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Q2)),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2))) ) ) ).

% div_less_iff_less_mult
tff(fact_4095_div__less__dividend,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),M2)) ) ) ).

% div_less_dividend
tff(fact_4096_div__eq__dividend__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = M2 )
      <=> ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_4097_div__if,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = zero_zero_nat ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = 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,M2),N)),N)) ) ) ) ).

% div_if
tff(fact_4098_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,M2: 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,N),Q2)),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q2))))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_4099_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),Q2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Q2)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_4100_dividend__less__times__div,axiom,
    ! [N: nat,M2: 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(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N))))) ) ).

% dividend_less_times_div
tff(fact_4101_dividend__less__div__times,axiom,
    ! [N: nat,M2: 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(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),N)))) ) ).

% dividend_less_div_times
tff(fact_4102_split__div,axiom,
    ! [P2: fun_nat_bool,M2: nat,N: nat] :
      ( pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P2,zero_zero_nat)) )
        & ( ( N != zero_zero_nat )
         => ! [I: nat,J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),N))
             => ( ( M2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I)),J) )
               => pp(aa_nat_bool(P2,I)) ) ) ) ) ) ).

% split_div
tff(fact_4103_power__diff__power__eq,axiom,
    ! [A: code_integer,N: nat,M2: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
         => ( divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M2),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
         => ( divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M2),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) = divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_4104_power__diff__power__eq,axiom,
    ! [A: nat,N: nat,M2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = 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,N),M2))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_4105_power__diff__power__eq,axiom,
    ! [A: int,N: nat,M2: nat] :
      ( ( A != zero_zero_int )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_4106_le__div__geq,axiom,
    ! [N: nat,M2: 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(ord_less_eq_nat,N),M2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = 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,M2),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_4107_split__div_H,axiom,
    ! [P2: fun_nat_bool,M2: nat,N: nat] :
      ( pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)))
    <=> ( ( ( N = zero_zero_nat )
          & pp(aa_nat_bool(P2,zero_zero_nat)) )
        | ? [Q5: 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,N),Q5)),M2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q5))))
            & pp(aa_nat_bool(P2,Q5)) ) ) ) ).

% split_div'
tff(fact_4108_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => 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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% div_2_gt_zero
tff(fact_4109_Suc__n__div__2__gt__zero,axiom,
    ! [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(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_4110_linear__plus__1__le__power,axiom,
    ! [X: real,N: 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(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(X),one_one_real)),N))) ) ).

% linear_plus_1_le_power
tff(fact_4111_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_4112_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_4113_set__bit__0,axiom,
    ! [A: code_integer] : bit_se2793503036327961859nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_4114_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_4115_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(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_4116_case4_I11_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(treeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na))) )
          & ! [X5: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),na) = I4 )
                & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),na))) )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,mi),X5))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),ma)) ) ) ) ) ) ).

% case4(11)
tff(fact_4117_enat__ord__number_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M2),numera1916890842035813515d_enat(N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N))) ) ).

% enat_ord_number(1)
tff(fact_4118_le__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z2))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Z2))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),Z2)) ) ) ).

% le_sup_iff
tff(fact_4119_le__sup__iff,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)),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,Y),Z2)) ) ) ).

% le_sup_iff
tff(fact_4120_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),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,Y),Z2)) ) ) ).

% le_sup_iff
tff(fact_4121_le__sup__iff,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),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,Y),Z2)) ) ) ).

% le_sup_iff
tff(fact_4122_sup_Obounded__iff,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)),A))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(C2),A)) ) ) ).

% sup.bounded_iff
tff(fact_4123_sup_Obounded__iff,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C2)),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,C2),A)) ) ) ).

% sup.bounded_iff
tff(fact_4124_sup_Obounded__iff,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C2)),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,C2),A)) ) ) ).

% sup.bounded_iff
tff(fact_4125_sup_Obounded__iff,axiom,
    ! [B: int,C2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C2)),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,C2),A)) ) ) ).

% sup.bounded_iff
tff(fact_4126_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_4127_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_4128_high__def,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = 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))),N)) ).

% high_def
tff(fact_4129_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M2: 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,N),M2))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))) ) ).

% high_bound_aux
tff(fact_4130_high__inv,axiom,
    ! [X: nat,N: 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))),N)))
     => ( 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))),N))),X)),N) = Y ) ) ).

% high_inv
tff(fact_4131_low__inv,axiom,
    ! [X: nat,N: 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))),N)))
     => ( 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))),N))),X)),N) = X ) ) ).

% low_inv
tff(fact_4132_le__inf__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(X,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z2))
    <=> ( ord_le3146513528884898305at_nat(X,Y)
        & ord_le3146513528884898305at_nat(X,Z2) ) ) ).

% le_inf_iff
tff(fact_4133_le__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z2)))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Z2)) ) ) ).

% le_inf_iff
tff(fact_4134_le__inf__iff,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z2)))
    <=> ( 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,X),Z2)) ) ) ).

% le_inf_iff
tff(fact_4135_le__inf__iff,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z2)))
    <=> ( 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,X),Z2)) ) ) ).

% le_inf_iff
tff(fact_4136_le__inf__iff,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z2)))
    <=> ( 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,X),Z2)) ) ) ).

% le_inf_iff
tff(fact_4137_inf_Obounded__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2))
    <=> ( ord_le3146513528884898305at_nat(A,B)
        & ord_le3146513528884898305at_nat(A,C2) ) ) ).

% inf.bounded_iff
tff(fact_4138_inf_Obounded__iff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2)))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C2)) ) ) ).

% inf.bounded_iff
tff(fact_4139_inf_Obounded__iff,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C2)))
    <=> ( 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,A),C2)) ) ) ).

% inf.bounded_iff
tff(fact_4140_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C2)))
    <=> ( 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),C2)) ) ) ).

% inf.bounded_iff
tff(fact_4141_inf_Obounded__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C2)))
    <=> ( 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,A),C2)) ) ) ).

% inf.bounded_iff
tff(fact_4142_both__member__options__ding,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary)),N))
     => ( 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_4143_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_4144_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_4145_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_4146_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M2: 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,N),M2))))
     => ( 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(ord_less_nat,zero_zero_nat),M2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_4147_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M2: 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,N),M2))))
     => ( 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(ord_less_nat,zero_zero_nat),M2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_4148_inf__sup__ord_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y),Y) ).

% inf_sup_ord(2)
tff(fact_4149_inf__sup__ord_I2_J,axiom,
    ! [X: set_nat,Y: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y)) ).

% inf_sup_ord(2)
tff(fact_4150_inf__sup__ord_I2_J,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),Y)) ).

% inf_sup_ord(2)
tff(fact_4151_inf__sup__ord_I2_J,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(inf_inf_nat,X),Y)),Y)) ).

% inf_sup_ord(2)
tff(fact_4152_inf__sup__ord_I2_J,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(inf_inf_int,X),Y)),Y)) ).

% inf_sup_ord(2)
tff(fact_4153_inf__sup__ord_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y),X) ).

% inf_sup_ord(1)
tff(fact_4154_inf__sup__ord_I1_J,axiom,
    ! [X: set_nat,Y: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),X)) ).

% inf_sup_ord(1)
tff(fact_4155_inf__sup__ord_I1_J,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),X)) ).

% inf_sup_ord(1)
tff(fact_4156_inf__sup__ord_I1_J,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(inf_inf_nat,X),Y)),X)) ).

% inf_sup_ord(1)
tff(fact_4157_inf__sup__ord_I1_J,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(inf_inf_int,X),Y)),X)) ).

% inf_sup_ord(1)
tff(fact_4158_inf__le1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y),X) ).

% inf_le1
tff(fact_4159_inf__le1,axiom,
    ! [X: set_nat,Y: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),X)) ).

% inf_le1
tff(fact_4160_inf__le1,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),X)) ).

% inf_le1
tff(fact_4161_inf__le1,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(inf_inf_nat,X),Y)),X)) ).

% inf_le1
tff(fact_4162_inf__le1,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(inf_inf_int,X),Y)),X)) ).

% inf_le1
tff(fact_4163_inf__le2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y),Y) ).

% inf_le2
tff(fact_4164_inf__le2,axiom,
    ! [X: set_nat,Y: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y)) ).

% inf_le2
tff(fact_4165_inf__le2,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),Y)) ).

% inf_le2
tff(fact_4166_inf__le2,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(inf_inf_nat,X),Y)),Y)) ).

% inf_le2
tff(fact_4167_inf__le2,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(inf_inf_int,X),Y)),Y)) ).

% inf_le2
tff(fact_4168_le__infE,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(X,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B))
     => ~ ( ord_le3146513528884898305at_nat(X,A)
         => ~ ord_le3146513528884898305at_nat(X,B) ) ) ).

% le_infE
tff(fact_4169_le__infE,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)))
     => ~ ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),A))
         => ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(X),B)) ) ) ).

% le_infE
tff(fact_4170_le__infE,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)))
     => ~ ( 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,X),B)) ) ) ).

% le_infE
tff(fact_4171_le__infE,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),A))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),B)) ) ) ).

% le_infE
tff(fact_4172_le__infE,axiom,
    ! [X: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)))
     => ~ ( 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,X),B)) ) ) ).

% le_infE
tff(fact_4173_le__infI,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(X,A)
     => ( ord_le3146513528884898305at_nat(X,B)
       => ord_le3146513528884898305at_nat(X,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) ) ) ).

% le_infI
tff(fact_4174_le__infI,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),A))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),B))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B))) ) ) ).

% le_infI
tff(fact_4175_le__infI,axiom,
    ! [X: rat,A: rat,B: 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,X),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B))) ) ) ).

% le_infI
tff(fact_4176_le__infI,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B))) ) ) ).

% le_infI
tff(fact_4177_le__infI,axiom,
    ! [X: int,A: int,B: 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,X),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B))) ) ) ).

% le_infI
tff(fact_4178_inf__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,C2)
     => ( ord_le3146513528884898305at_nat(B,D)
       => ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),D)) ) ) ).

% inf_mono
tff(fact_4179_inf__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C2))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),D))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),D))) ) ) ).

% inf_mono
tff(fact_4180_inf__mono,axiom,
    ! [A: rat,C2: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C2))
     => ( 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_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,C2),D))) ) ) ).

% inf_mono
tff(fact_4181_inf__mono,axiom,
    ! [A: nat,C2: nat,B: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C2))
     => ( 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_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,C2),D))) ) ) ).

% inf_mono
tff(fact_4182_inf__mono,axiom,
    ! [A: int,C2: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C2))
     => ( 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_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),aa_int_int(aa_int_fun_int_int(inf_inf_int,C2),D))) ) ) ).

% inf_mono
tff(fact_4183_le__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,X)
     => ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B),X) ) ).

% le_infI1
tff(fact_4184_le__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),X))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X)) ) ).

% le_infI1
tff(fact_4185_le__infI1,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),X)) ) ).

% le_infI1
tff(fact_4186_le__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X)) ) ).

% le_infI1
tff(fact_4187_le__infI1,axiom,
    ! [A: int,X: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X)) ) ).

% le_infI1
tff(fact_4188_le__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(B,X)
     => ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B),X) ) ).

% le_infI2
tff(fact_4189_le__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),X))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X)) ) ).

% le_infI2
tff(fact_4190_le__infI2,axiom,
    ! [B: rat,X: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),X)) ) ).

% le_infI2
tff(fact_4191_le__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X)) ) ).

% le_infI2
tff(fact_4192_le__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X)) ) ).

% le_infI2
tff(fact_4193_inf_OorderE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,B)
     => ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ) ).

% inf.orderE
tff(fact_4194_inf_OorderE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).

% inf.orderE
tff(fact_4195_inf_OorderE,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) ) ) ).

% inf.orderE
tff(fact_4196_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ) ).

% inf.orderE
tff(fact_4197_inf_OorderE,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ) ) ).

% inf.orderE
tff(fact_4198_inf_OorderI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) )
     => ord_le3146513528884898305at_nat(A,B) ) ).

% inf.orderI
tff(fact_4199_inf_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ).

% inf.orderI
tff(fact_4200_inf_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% inf.orderI
tff(fact_4201_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% inf.orderI
tff(fact_4202_inf_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% inf.orderI
tff(fact_4203_inf__unique,axiom,
    ! [F2: fun_se3560235952666129655at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F2,X3),Y3),X3)
     => ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F2,X3),Y3),Y3)
       => ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
              ( ord_le3146513528884898305at_nat(X3,Y3)
             => ( ord_le3146513528884898305at_nat(X3,Z)
               => ord_le3146513528884898305at_nat(X3,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F2,Y3),Z)) ) )
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F2,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_4204_inf__unique,axiom,
    ! [F2: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F2,X3),Y3)),X3))
     => ( ! [X3: set_nat,Y3: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F2,X3),Y3)),Y3))
       => ( ! [X3: set_nat,Y3: set_nat,Z: set_nat] :
              ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X3),Y3))
             => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X3),Z))
               => pp(aa_set_nat_bool(ord_less_eq_set_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F2,Y3),Z))) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F2,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_4205_inf__unique,axiom,
    ! [F2: fun_rat_fun_rat_rat,X: rat,Y: rat] :
      ( ! [X3: rat,Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F2,X3),Y3)),X3))
     => ( ! [X3: rat,Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F2,X3),Y3)),Y3))
       => ( ! [X3: rat,Y3: rat,Z: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y3))
             => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Z))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(F2,Y3),Z))) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(F2,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_4206_inf__unique,axiom,
    ! [F2: fun_nat_fun_nat_nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F2,X3),Y3)),X3))
     => ( ! [X3: nat,Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F2,X3),Y3)),Y3))
       => ( ! [X3: nat,Y3: nat,Z: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Z))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(F2,Y3),Z))) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F2,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_4207_inf__unique,axiom,
    ! [F2: fun_int_fun_int_int,X: int,Y: int] :
      ( ! [X3: int,Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F2,X3),Y3)),X3))
     => ( ! [X3: int,Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F2,X3),Y3)),Y3))
       => ( ! [X3: int,Y3: int,Z: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y3))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Z))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(F2,Y3),Z))) ) )
         => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = aa_int_int(aa_int_fun_int_int(F2,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_4208_le__iff__inf,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(X,Y)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_4209_le__iff__inf,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_4210_le__iff__inf,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_4211_le__iff__inf,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(inf_inf_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_4212_le__iff__inf,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(inf_inf_int,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_4213_inf_Oabsorb1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,B)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_4214_inf_Oabsorb1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_4215_inf_Oabsorb1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_4216_inf_Oabsorb1,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(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_4217_inf_Oabsorb1,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(inf_inf_int,A),B) = A ) ) ).

% inf.absorb1
tff(fact_4218_inf_Oabsorb2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(B,A)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_4219_inf_Oabsorb2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_4220_inf_Oabsorb2,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_4221_inf_Oabsorb2,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(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_4222_inf_Oabsorb2,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(inf_inf_int,A),B) = B ) ) ).

% inf.absorb2
tff(fact_4223_inf__absorb1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(X,Y)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_4224_inf__absorb1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_4225_inf__absorb1,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_4226_inf__absorb1,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(inf_inf_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_4227_inf__absorb1,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(inf_inf_int,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_4228_inf__absorb2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(Y,X)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_4229_inf__absorb2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_4230_inf__absorb2,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_4231_inf__absorb2,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(inf_inf_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_4232_inf__absorb2,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(inf_inf_int,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_4233_inf_OboundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2))
     => ~ ( ord_le3146513528884898305at_nat(A,B)
         => ~ ord_le3146513528884898305at_nat(A,C2) ) ) ).

% inf.boundedE
tff(fact_4234_inf_OboundedE,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2)))
     => ~ ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
         => ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C2)) ) ) ).

% inf.boundedE
tff(fact_4235_inf_OboundedE,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C2)))
     => ~ ( 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,A),C2)) ) ) ).

% inf.boundedE
tff(fact_4236_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C2)))
     => ~ ( 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),C2)) ) ) ).

% inf.boundedE
tff(fact_4237_inf_OboundedE,axiom,
    ! [A: int,B: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C2)))
     => ~ ( 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,A),C2)) ) ) ).

% inf.boundedE
tff(fact_4238_inf_OboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,B)
     => ( ord_le3146513528884898305at_nat(A,C2)
       => ord_le3146513528884898305at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2)) ) ) ).

% inf.boundedI
tff(fact_4239_inf_OboundedI,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C2))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2))) ) ) ).

% inf.boundedI
tff(fact_4240_inf_OboundedI,axiom,
    ! [A: rat,B: rat,C2: 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,A),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C2))) ) ) ).

% inf.boundedI
tff(fact_4241_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C2))) ) ) ).

% inf.boundedI
tff(fact_4242_inf_OboundedI,axiom,
    ! [A: int,B: int,C2: 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,A),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C2))) ) ) ).

% inf.boundedI
tff(fact_4243_inf__greatest,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(X,Y)
     => ( ord_le3146513528884898305at_nat(X,Z2)
       => ord_le3146513528884898305at_nat(X,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z2)) ) ) ).

% inf_greatest
tff(fact_4244_inf__greatest,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(X),Z2))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z2))) ) ) ).

% inf_greatest
tff(fact_4245_inf__greatest,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,X),Z2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z2))) ) ) ).

% inf_greatest
tff(fact_4246_inf__greatest,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,X),Z2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z2))) ) ) ).

% inf_greatest
tff(fact_4247_inf__greatest,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,X),Z2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z2))) ) ) ).

% inf_greatest
tff(fact_4248_inf_Oorder__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,B)
    <=> ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_4249_inf_Oorder__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
    <=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_4250_inf_Oorder__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) ) ) ).

% inf.order_iff
tff(fact_4251_inf_Oorder__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_4252_inf_Oorder__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ) ) ).

% inf.order_iff
tff(fact_4253_inf_Ocobounded1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B),A) ).

% inf.cobounded1
tff(fact_4254_inf_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),A)) ).

% inf.cobounded1
tff(fact_4255_inf_Ocobounded1,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),A)) ).

% inf.cobounded1
tff(fact_4256_inf_Ocobounded1,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(inf_inf_nat,A),B)),A)) ).

% inf.cobounded1
tff(fact_4257_inf_Ocobounded1,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(inf_inf_int,A),B)),A)) ).

% inf.cobounded1
tff(fact_4258_inf_Ocobounded2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B),B) ).

% inf.cobounded2
tff(fact_4259_inf_Ocobounded2,axiom,
    ! [A: set_nat,B: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),B)) ).

% inf.cobounded2
tff(fact_4260_inf_Ocobounded2,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),B)) ).

% inf.cobounded2
tff(fact_4261_inf_Ocobounded2,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(inf_inf_nat,A),B)),B)) ).

% inf.cobounded2
tff(fact_4262_inf_Ocobounded2,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(inf_inf_int,A),B)),B)) ).

% inf.cobounded2
tff(fact_4263_inf_Oabsorb__iff1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,B)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_4264_inf_Oabsorb__iff1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_4265_inf_Oabsorb__iff1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_4266_inf_Oabsorb__iff1,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(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_4267_inf_Oabsorb__iff1,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(inf_inf_int,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_4268_inf_Oabsorb__iff2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(B,A)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_4269_inf_Oabsorb__iff2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_4270_inf_Oabsorb__iff2,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_4271_inf_Oabsorb__iff2,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(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_4272_inf_Oabsorb__iff2,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(inf_inf_int,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_4273_inf_OcoboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A,C2)
     => ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B),C2) ) ).

% inf.coboundedI1
tff(fact_4274_inf_OcoboundedI1,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C2))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C2)) ) ).

% inf.coboundedI1
tff(fact_4275_inf_OcoboundedI1,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C2)) ) ).

% inf.coboundedI1
tff(fact_4276_inf_OcoboundedI1,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C2)) ) ).

% inf.coboundedI1
tff(fact_4277_inf_OcoboundedI1,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C2)) ) ).

% inf.coboundedI1
tff(fact_4278_inf_OcoboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(B,C2)
     => ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B),C2) ) ).

% inf.coboundedI2
tff(fact_4279_inf_OcoboundedI2,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),C2))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C2)) ) ).

% inf.coboundedI2
tff(fact_4280_inf_OcoboundedI2,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C2)) ) ).

% inf.coboundedI2
tff(fact_4281_inf_OcoboundedI2,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C2)) ) ).

% inf.coboundedI2
tff(fact_4282_inf_OcoboundedI2,axiom,
    ! [B: int,C2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C2)) ) ).

% inf.coboundedI2
tff(fact_4283_sup_OcoboundedI2,axiom,
    ! [C2: set_nat,B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C2),B))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ) ).

% sup.coboundedI2
tff(fact_4284_sup_OcoboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ) ).

% sup.coboundedI2
tff(fact_4285_sup_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B))) ) ).

% sup.coboundedI2
tff(fact_4286_sup_OcoboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B))) ) ).

% sup.coboundedI2
tff(fact_4287_sup_OcoboundedI1,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C2),A))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ) ).

% sup.coboundedI1
tff(fact_4288_sup_OcoboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ) ).

% sup.coboundedI1
tff(fact_4289_sup_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B))) ) ).

% sup.coboundedI1
tff(fact_4290_sup_OcoboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B))) ) ).

% sup.coboundedI1
tff(fact_4291_sup_Oabsorb__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_4292_sup_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(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_4293_sup_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(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_4294_sup_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(sup_sup_int,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_4295_sup_Oabsorb__iff1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_4296_sup_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(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_4297_sup_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(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_4298_sup_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(sup_sup_int,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_4299_sup_Ocobounded2,axiom,
    ! [B: set_nat,A: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ).

% sup.cobounded2
tff(fact_4300_sup_Ocobounded2,axiom,
    ! [B: rat,A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ).

% sup.cobounded2
tff(fact_4301_sup_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(sup_sup_nat,A),B))) ).

% sup.cobounded2
tff(fact_4302_sup_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(sup_sup_int,A),B))) ).

% sup.cobounded2
tff(fact_4303_sup_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ).

% sup.cobounded1
tff(fact_4304_sup_Ocobounded1,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ).

% sup.cobounded1
tff(fact_4305_sup_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(sup_sup_nat,A),B))) ).

% sup.cobounded1
tff(fact_4306_sup_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(sup_sup_int,A),B))) ).

% sup.cobounded1
tff(fact_4307_sup_Oorder__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
    <=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_4308_sup_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(aa_rat_fun_rat_rat(sup_sup_rat,A),B) ) ) ).

% sup.order_iff
tff(fact_4309_sup_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(sup_sup_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_4310_sup_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(sup_sup_int,A),B) ) ) ).

% sup.order_iff
tff(fact_4311_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C2),A))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)),A)) ) ) ).

% sup.boundedI
tff(fact_4312_sup_OboundedI,axiom,
    ! [B: rat,A: rat,C2: 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,C2),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C2)),A)) ) ) ).

% sup.boundedI
tff(fact_4313_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C2: 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,C2),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C2)),A)) ) ) ).

% sup.boundedI
tff(fact_4314_sup_OboundedI,axiom,
    ! [B: int,A: int,C2: 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,C2),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C2)),A)) ) ) ).

% sup.boundedI
tff(fact_4315_sup_OboundedE,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)),A))
     => ~ ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
         => ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(C2),A)) ) ) ).

% sup.boundedE
tff(fact_4316_sup_OboundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C2)),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,C2),A)) ) ) ).

% sup.boundedE
tff(fact_4317_sup_OboundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C2)),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,C2),A)) ) ) ).

% sup.boundedE
tff(fact_4318_sup_OboundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C2)),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,C2),A)) ) ) ).

% sup.boundedE
tff(fact_4319_sup__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_4320_sup__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_4321_sup__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(sup_sup_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_4322_sup__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(sup_sup_int,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_4323_sup__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_4324_sup__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_4325_sup__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(sup_sup_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_4326_sup__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(sup_sup_int,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_4327_sup_Oabsorb2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_4328_sup_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_4329_sup_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(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_4330_sup_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(sup_sup_int,A),B) = B ) ) ).

% sup.absorb2
tff(fact_4331_sup_Oabsorb1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_4332_sup_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_4333_sup_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(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_4334_sup_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(sup_sup_int,A),B) = A ) ) ).

% sup.absorb1
tff(fact_4335_sup__unique,axiom,
    ! [F2: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F2,X3),Y3)))
     => ( ! [X3: set_nat,Y3: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(Y3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F2,X3),Y3)))
       => ( ! [X3: set_nat,Y3: set_nat,Z: set_nat] :
              ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y3),X3))
             => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Z),X3))
               => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F2,Y3),Z)),X3)) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F2,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_4336_sup__unique,axiom,
    ! [F2: fun_rat_fun_rat_rat,X: rat,Y: rat] :
      ( ! [X3: rat,Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(F2,X3),Y3)))
     => ( ! [X3: rat,Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),aa_rat_rat(aa_rat_fun_rat_rat(F2,X3),Y3)))
       => ( ! [X3: rat,Y3: rat,Z: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X3))
             => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F2,Y3),Z)),X3)) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(F2,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_4337_sup__unique,axiom,
    ! [F2: fun_nat_fun_nat_nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(F2,X3),Y3)))
     => ( ! [X3: nat,Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(F2,X3),Y3)))
       => ( ! [X3: nat,Y3: nat,Z: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F2,Y3),Z)),X3)) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F2,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_4338_sup__unique,axiom,
    ! [F2: fun_int_fun_int_int,X: int,Y: int] :
      ( ! [X3: int,Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(F2,X3),Y3)))
     => ( ! [X3: int,Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),aa_int_int(aa_int_fun_int_int(F2,X3),Y3)))
       => ( ! [X3: int,Y3: int,Z: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X3))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X3))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F2,Y3),Z)),X3)) ) )
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = aa_int_int(aa_int_fun_int_int(F2,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_4339_sup_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A)) ) ).

% sup.orderI
tff(fact_4340_sup_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% sup.orderI
tff(fact_4341_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% sup.orderI
tff(fact_4342_sup_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% sup.orderI
tff(fact_4343_sup_OorderE,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).

% sup.orderE
tff(fact_4344_sup_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(aa_rat_fun_rat_rat(sup_sup_rat,A),B) ) ) ).

% sup.orderE
tff(fact_4345_sup_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(sup_sup_nat,A),B) ) ) ).

% sup.orderE
tff(fact_4346_sup_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(sup_sup_int,A),B) ) ) ).

% sup.orderE
tff(fact_4347_le__iff__sup,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_4348_le__iff__sup,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_4349_le__iff__sup,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(sup_sup_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_4350_le__iff__sup,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(sup_sup_int,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_4351_sup__least,axiom,
    ! [Y: set_nat,X: set_nat,Z2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Z2),X))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z2)),X)) ) ) ).

% sup_least
tff(fact_4352_sup__least,axiom,
    ! [Y: rat,X: rat,Z2: 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,Z2),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Y),Z2)),X)) ) ) ).

% sup_least
tff(fact_4353_sup__least,axiom,
    ! [Y: nat,X: nat,Z2: 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,Z2),X))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z2)),X)) ) ) ).

% sup_least
tff(fact_4354_sup__least,axiom,
    ! [Y: int,X: int,Z2: 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,Z2),X))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z2)),X)) ) ) ).

% sup_least
tff(fact_4355_sup__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C2))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),D))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),D))) ) ) ).

% sup_mono
tff(fact_4356_sup__mono,axiom,
    ! [A: rat,C2: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C2))
     => ( 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_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C2),D))) ) ) ).

% sup_mono
tff(fact_4357_sup__mono,axiom,
    ! [A: nat,C2: nat,B: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C2))
     => ( 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_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C2),D))) ) ) ).

% sup_mono
tff(fact_4358_sup__mono,axiom,
    ! [A: int,C2: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C2))
     => ( 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_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),aa_int_int(aa_int_fun_int_int(sup_sup_int,C2),D))) ) ) ).

% sup_mono
tff(fact_4359_sup_Omono,axiom,
    ! [C2: set_nat,A: set_nat,D: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C2),A))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(D),B))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),D)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ) ) ).

% sup.mono
tff(fact_4360_sup_Omono,axiom,
    ! [C2: rat,A: rat,D: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),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(aa_rat_fun_rat_rat(sup_sup_rat,C2),D)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ) ) ).

% sup.mono
tff(fact_4361_sup_Omono,axiom,
    ! [C2: nat,A: nat,D: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),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(sup_sup_nat,C2),D)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B))) ) ) ).

% sup.mono
tff(fact_4362_sup_Omono,axiom,
    ! [C2: int,A: int,D: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),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(sup_sup_int,C2),D)),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B))) ) ) ).

% sup.mono
tff(fact_4363_le__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),B))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ) ).

% le_supI2
tff(fact_4364_le__supI2,axiom,
    ! [X: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ) ).

% le_supI2
tff(fact_4365_le__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B))) ) ).

% le_supI2
tff(fact_4366_le__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B))) ) ).

% le_supI2
tff(fact_4367_le__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),A))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ) ).

% le_supI1
tff(fact_4368_le__supI1,axiom,
    ! [X: rat,A: rat,B: 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,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ) ).

% le_supI1
tff(fact_4369_le__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B))) ) ).

% le_supI1
tff(fact_4370_le__supI1,axiom,
    ! [X: int,A: int,B: 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,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B))) ) ).

% le_supI1
tff(fact_4371_sup__ge2,axiom,
    ! [Y: set_nat,X: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y))) ).

% sup_ge2
tff(fact_4372_sup__ge2,axiom,
    ! [Y: rat,X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y))) ).

% sup_ge2
tff(fact_4373_sup__ge2,axiom,
    ! [Y: nat,X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y))) ).

% sup_ge2
tff(fact_4374_sup__ge2,axiom,
    ! [Y: int,X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y))) ).

% sup_ge2
tff(fact_4375_sup__ge1,axiom,
    ! [X: set_nat,Y: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y))) ).

% sup_ge1
tff(fact_4376_sup__ge1,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y))) ).

% sup_ge1
tff(fact_4377_sup__ge1,axiom,
    ! [X: nat,Y: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y))) ).

% sup_ge1
tff(fact_4378_sup__ge1,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y))) ).

% sup_ge1
tff(fact_4379_le__supI,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),X))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),X))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),X)) ) ) ).

% le_supI
tff(fact_4380_le__supI,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),X)) ) ) ).

% le_supI
tff(fact_4381_le__supI,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),X))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),X)) ) ) ).

% le_supI
tff(fact_4382_le__supI,axiom,
    ! [A: int,X: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),X))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),X)) ) ) ).

% le_supI
tff(fact_4383_le__supE,axiom,
    ! [A: set_nat,B: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),X))
     => ~ ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),X))
         => ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(B),X)) ) ) ).

% le_supE
tff(fact_4384_le__supE,axiom,
    ! [A: rat,B: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),X))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),X))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),X)) ) ) ).

% le_supE
tff(fact_4385_le__supE,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),X))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),X)) ) ) ).

% le_supE
tff(fact_4386_le__supE,axiom,
    ! [A: int,B: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),X))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),X)) ) ) ).

% le_supE
tff(fact_4387_inf__sup__ord_I3_J,axiom,
    ! [X: set_nat,Y: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y))) ).

% inf_sup_ord(3)
tff(fact_4388_inf__sup__ord_I3_J,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y))) ).

% inf_sup_ord(3)
tff(fact_4389_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y))) ).

% inf_sup_ord(3)
tff(fact_4390_inf__sup__ord_I3_J,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y))) ).

% inf_sup_ord(3)
tff(fact_4391_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,X: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y))) ).

% inf_sup_ord(4)
tff(fact_4392_inf__sup__ord_I4_J,axiom,
    ! [Y: rat,X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y))) ).

% inf_sup_ord(4)
tff(fact_4393_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y))) ).

% inf_sup_ord(4)
tff(fact_4394_inf__sup__ord_I4_J,axiom,
    ! [Y: int,X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y))) ).

% inf_sup_ord(4)
tff(fact_4395_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M2))
       => ( ( 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))),M2) )
         => ( ( M2 = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
             => ( ! [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))),M2)))
                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),X_13))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) ) )
                 => ( 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 )
                         => ! [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))),M2)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I3 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
                                & ! [X3: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N) = I3 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X3))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),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_4396_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M2))
       => ( ( 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))),M2) )
         => ( ( M2 = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
             => ( ! [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))),M2)))
                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),X_13))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) ) )
                 => ( 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 )
                         => ! [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))),M2)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I3 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
                                & ! [X3: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N) = I3 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X3))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),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_4397_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A12),A23))
     => ( ( ? [A3: bool,B3: bool] : A12 = vEBT_Leaf(A3,B3)
         => ( A23 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
              ( ( A12 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2) )
             => ( ( A23 = Deg2 )
               => ( ! [X5: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
                     => pp(aa_nat_bool(vEBT_invar_vebt(X5),N3)) )
                 => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M))
                   => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                     => ( ( M = N3 )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M) )
                         => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_1))
                           => ~ ! [X5: vEBT_VEBT] :
                                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
                                 => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_1)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
                ( ( A12 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2) )
               => ( ( A23 = Deg2 )
                 => ( ! [X5: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
                       => pp(aa_nat_bool(vEBT_invar_vebt(X5),N3)) )
                   => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M))
                     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                       => ( ( M = aa_nat_nat(suc,N3) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M) )
                           => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_1))
                             => ~ ! [X5: vEBT_VEBT] :
                                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
                                   => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_1)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList,Summary2) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X5: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
                         => pp(aa_nat_bool(vEBT_invar_vebt(X5),N3)) )
                     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M))
                       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                         => ( ( M = N3 )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M) )
                             => ( ! [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,Summary2),I4)) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X5: vEBT_VEBT] :
                                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
                                       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_1)) ) )
                                 => ( 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))),M)))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = 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),N3))) )
                                                & ! [X5: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N3) = 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,X5),N3))) )
                                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X5))
                                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList,Summary2) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X5: vEBT_VEBT] :
                            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
                           => pp(aa_nat_bool(vEBT_invar_vebt(X5),N3)) )
                       => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M))
                         => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                           => ( ( M = aa_nat_nat(suc,N3) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M) )
                               => ( ! [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,Summary2),I4)) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X5: vEBT_VEBT] :
                                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
                                         => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_1)) ) )
                                   => ( 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))),M)))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = 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),N3))) )
                                                  & ! [X5: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N3) = 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,X5),N3))) )
                                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X5))
                                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_4398_invar__vebt_Osimps,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A12),A23))
    <=> ( ( ? [A5: bool,B4: bool] : A12 = vEBT_Leaf(A5,B4)
          & ( A23 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList4: list_VEBT_VEBT,N4: nat,Summary4: vEBT_VEBT] :
            ( ( A12 = vEBT_Node(none_P5556105721700978146at_nat,A23,TreeList4,Summary4) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList4)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X4),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary4),N4))
            & ( size_s6755466524823107622T_VEBT(TreeList4) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4) )
            & ( A23 = 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,Summary4),X_13))
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList4)))
               => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_13)) ) )
        | ? [TreeList4: list_VEBT_VEBT,N4: nat,Summary4: vEBT_VEBT] :
            ( ( A12 = vEBT_Node(none_P5556105721700978146at_nat,A23,TreeList4,Summary4) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList4)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X4),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary4),aa_nat_nat(suc,N4)))
            & ( size_s6755466524823107622T_VEBT(TreeList4) = aa_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)) )
            & ( A23 = 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,Summary4),X_13))
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList4)))
               => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_13)) ) )
        | ? [TreeList4: list_VEBT_VEBT,N4: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A12 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A23,TreeList4,Summary4) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList4)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X4),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary4),N4))
            & ( size_s6755466524823107622T_VEBT(TreeList4) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4) )
            & ( A23 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ! [I: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)))
               => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList4,I)),X_13))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary4),I)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList4)))
                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),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))),A23)))
            & ( ( Mi3 != Ma3 )
             => ! [I: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N4) = I )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList4,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N4))) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N4) = I )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList4,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N4))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X4))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma3)) ) ) ) ) ) )
        | ? [TreeList4: list_VEBT_VEBT,N4: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A12 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A23,TreeList4,Summary4) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList4)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X4),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary4),aa_nat_nat(suc,N4)))
            & ( size_s6755466524823107622T_VEBT(TreeList4) = aa_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)) )
            & ( A23 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ! [I: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4))))
               => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList4,I)),X_13))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary4),I)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList4)))
                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),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))),A23)))
            & ( ( Mi3 != Ma3 )
             => ! [I: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4))))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N4) = I )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList4,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N4))) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N4) = I )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList4,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N4))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X4))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_4399_distrib__inf__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z2))) ).

% distrib_inf_le
tff(fact_4400_distrib__inf__le,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z2))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z2)))) ).

% distrib_inf_le
tff(fact_4401_distrib__inf__le,axiom,
    ! [X: rat,Y: rat,Z2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Z2))),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Y),Z2)))) ).

% distrib_inf_le
tff(fact_4402_distrib__inf__le,axiom,
    ! [X: nat,Y: nat,Z2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z2))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z2)))) ).

% distrib_inf_le
tff(fact_4403_distrib__inf__le,axiom,
    ! [X: int,Y: int,Z2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Z2))),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z2)))) ).

% distrib_inf_le
tff(fact_4404_distrib__sup__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] : ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z2))) ).

% distrib_sup_le
tff(fact_4405_distrib__sup__le,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z2))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z2)))) ).

% distrib_sup_le
tff(fact_4406_distrib__sup__le,axiom,
    ! [X: rat,Y: rat,Z2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z2))),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Z2)))) ).

% distrib_sup_le
tff(fact_4407_distrib__sup__le,axiom,
    ! [X: nat,Y: nat,Z2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z2))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z2)))) ).

% distrib_sup_le
tff(fact_4408_distrib__sup__le,axiom,
    ! [X: int,Y: int,Z2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z2))),aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z2)))) ).

% distrib_sup_le
tff(fact_4409_in__children__def,axiom,
    ! [N: nat,TreeList2: list_VEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(N,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),N))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N))) ) ).

% in_children_def
tff(fact_4410_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va3: 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)),N))
     => ( ( N = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
       => ( ~ 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,Va3),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,Va3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ).

% nested_mint
tff(fact_4411_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_4412_set__encode__insert,axiom,
    ! [A4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ~ pp(aa_set_nat_bool(member_nat(N),A4))
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(N,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_set_nat_nat(nat_set_encode,A4)) ) ) ) ).

% set_encode_insert
tff(fact_4413_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_4414_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(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_4415_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_4416_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_4417_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),product_Pair_int_int(Q2,R)))
       => pp(aa_Pro4501881432599481050t_bool(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(Q2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R)),one_one_int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_4418_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,X)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_4419_length__list__update,axiom,
    ! [Xs: list_bool,I2: nat,X: bool] : size_size_list_bool(list_update_bool(Xs,I2,X)) = size_size_list_bool(Xs) ).

% length_list_update
tff(fact_4420_length__list__update,axiom,
    ! [Xs: list_nat,I2: nat,X: nat] : aa_list_nat_nat(size_size_list_nat,list_update_nat(Xs,I2,X)) = aa_list_nat_nat(size_size_list_nat,Xs) ).

% length_list_update
tff(fact_4421_length__list__update,axiom,
    ! [Xs: list_int,I2: nat,X: int] : size_size_list_int(list_update_int(Xs,I2,X)) = size_size_list_int(Xs) ).

% length_list_update
tff(fact_4422_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_4423_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( some_nat(the_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_4424_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( some_num(the_num(Option)) = Option ) ) ).

% option.collapse
tff(fact_4425_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I2))
     => ( list_u1324408373059187874T_VEBT(Xs,I2,X) = Xs ) ) ).

% list_update_beyond
tff(fact_4426_list__update__beyond,axiom,
    ! [Xs: list_bool,I2: nat,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),I2))
     => ( list_update_bool(Xs,I2,X) = Xs ) ) ).

% list_update_beyond
tff(fact_4427_list__update__beyond,axiom,
    ! [Xs: list_nat,I2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),I2))
     => ( list_update_nat(Xs,I2,X) = Xs ) ) ).

% list_update_beyond
tff(fact_4428_list__update__beyond,axiom,
    ! [Xs: list_int,I2: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),I2))
     => ( list_update_int(Xs,I2,X) = Xs ) ) ).

% list_update_beyond
tff(fact_4429_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_4430_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,X),I2) = X ) ) ).

% nth_list_update_eq
tff(fact_4431_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I2,X)),I2))
      <=> pp(X) ) ) ).

% nth_list_update_eq
tff(fact_4432_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I2,X)),I2) = X ) ) ).

% nth_list_update_eq
tff(fact_4433_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I2,X)),I2) = X ) ) ).

% nth_list_update_eq
tff(fact_4434_set__swap,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_s6755466524823107622T_VEBT(Xs)))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,nth_VEBT_VEBT(Xs,J2)),J2,nth_VEBT_VEBT(Xs,I2))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_4435_set__swap,axiom,
    ! [I2: nat,Xs: list_bool,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_bool(Xs)))
       => ( set_bool2(list_update_bool(list_update_bool(Xs,I2,aa_nat_bool(nth_bool(Xs),J2)),J2,aa_nat_bool(nth_bool(Xs),I2))) = set_bool2(Xs) ) ) ) ).

% set_swap
tff(fact_4436_set__swap,axiom,
    ! [I2: nat,Xs: list_nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I2,aa_nat_nat(nth_nat(Xs),J2)),J2,aa_nat_nat(nth_nat(Xs),I2))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_4437_set__swap,axiom,
    ! [I2: nat,Xs: list_int,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_int(Xs)))
       => ( set_int2(list_update_int(list_update_int(Xs,I2,aa_nat_int(nth_int(Xs),J2)),J2,aa_nat_int(nth_int(Xs),I2))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_4438_unique__quotient,axiom,
    ! [A: int,B: int,Q2: int,R: int,Q4: int,R4: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),product_Pair_int_int(Q2,R)))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),product_Pair_int_int(Q4,R4)))
       => ( Q2 = Q4 ) ) ) ).

% unique_quotient
tff(fact_4439_unique__remainder,axiom,
    ! [A: int,B: int,Q2: int,R: int,Q4: int,R4: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),product_Pair_int_int(Q2,R)))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),product_Pair_int_int(Q4,R4)))
       => ( R = R4 ) ) ) ).

% unique_remainder
tff(fact_4440_zip__update,axiom,
    ! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] : zip_VE537291747668921783T_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,X),list_u1324408373059187874T_VEBT(Ys,I2,Y)) = list_u6961636818849549845T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys),I2,produc537772716801021591T_VEBT(X,Y)) ).

% zip_update
tff(fact_4441_zip__update,axiom,
    ! [Xs: list_Code_integer,I2: nat,X: code_integer,Ys: list_Code_integer,Y: code_integer] : zip_Co3543743374963494515nteger(list_u5447711078246177391nteger(Xs,I2,X),list_u5447711078246177391nteger(Ys,I2,Y)) = list_u2254550707601501961nteger(zip_Co3543743374963494515nteger(Xs,Ys),I2,produc1086072967326762835nteger(X,Y)) ).

% zip_update
tff(fact_4442_zip__update,axiom,
    ! [Xs: list_P6011104703257516679at_nat,I2: nat,X: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] : zip_Pr4664179122662387191at_nat(list_u6180841689913720943at_nat(Xs,I2,X),list_u6180841689913720943at_nat(Ys,I2,Y)) = list_u5003261594476800725at_nat(zip_Pr4664179122662387191at_nat(Xs,Ys),I2,produc6161850002892822231at_nat(X,Y)) ).

% zip_update
tff(fact_4443_zip__update,axiom,
    ! [Xs: list_s1210847774152347623at_nat,I2: nat,X: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat] : zip_se5600341670672612855at_nat(list_u8444657558853818831at_nat(Xs,I2,X),list_u8444657558853818831at_nat(Ys,I2,Y)) = list_u4696772448584712917at_nat(zip_se5600341670672612855at_nat(Xs,Ys),I2,produc2922128104949294807at_nat(X,Y)) ).

% zip_update
tff(fact_4444_zip__update,axiom,
    ! [Xs: list_nat,I2: nat,X: nat,Ys: list_nat,Y: nat] : zip_nat_nat(list_update_nat(Xs,I2,X),list_update_nat(Ys,I2,Y)) = list_u6180841689913720943at_nat(zip_nat_nat(Xs,Ys),I2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)) ).

% zip_update
tff(fact_4445_zip__update,axiom,
    ! [Xs: list_int,I2: nat,X: int,Ys: list_int,Y: int] : zip_int_int(list_update_int(Xs,I2,X),list_update_int(Ys,I2,Y)) = list_u3002344382305578791nt_int(zip_int_int(Xs,Ys),I2,product_Pair_int_int(X,Y)) ).

% zip_update
tff(fact_4446_option_Osel,axiom,
    ! [X2: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X2)) = X2 ).

% option.sel
tff(fact_4447_option_Osel,axiom,
    ! [X2: nat] : the_nat(some_nat(X2)) = X2 ).

% option.sel
tff(fact_4448_option_Osel,axiom,
    ! [X2: num] : the_num(some_num(X2)) = X2 ).

% option.sel
tff(fact_4449_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
      <=> ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( the_nat(Option) = the_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_4450_option_Oexpand,axiom,
    ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
      ( ( ( Option = none_P5556105721700978146at_nat )
      <=> ( Option2 = none_P5556105721700978146at_nat ) )
     => ( ( ( Option != none_P5556105721700978146at_nat )
         => ( ( Option2 != none_P5556105721700978146at_nat )
           => ( the_Pr8591224930841456533at_nat(Option) = the_Pr8591224930841456533at_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_4451_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
      <=> ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( the_num(Option) = the_num(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_4452_eucl__rel__int__by0,axiom,
    ! [K2: int] : pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K2,zero_zero_int),product_Pair_int_int(zero_zero_int,K2))) ).

% eucl_rel_int_by0
tff(fact_4453_div__int__unique,axiom,
    ! [K2: int,L: int,Q2: int,R: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K2,L),product_Pair_int_int(Q2,R)))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = Q2 ) ) ).

% div_int_unique
tff(fact_4454_list__update__code_I3_J,axiom,
    ! [X: nat,Xs: list_nat,I2: nat,Y: nat] : list_update_nat(cons_nat(X,Xs),aa_nat_nat(suc,I2),Y) = cons_nat(X,list_update_nat(Xs,I2,Y)) ).

% list_update_code(3)
tff(fact_4455_list__update__code_I3_J,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,I2: nat,Y: vEBT_VEBT] : list_u1324408373059187874T_VEBT(cons_VEBT_VEBT(X,Xs),aa_nat_nat(suc,I2),Y) = cons_VEBT_VEBT(X,list_u1324408373059187874T_VEBT(Xs,I2,Y)) ).

% list_update_code(3)
tff(fact_4456_list__update__code_I2_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] : list_update_nat(cons_nat(X,Xs),zero_zero_nat,Y) = cons_nat(Y,Xs) ).

% list_update_code(2)
tff(fact_4457_list__update__code_I2_J,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: vEBT_VEBT] : list_u1324408373059187874T_VEBT(cons_VEBT_VEBT(X,Xs),zero_zero_nat,Y) = cons_VEBT_VEBT(Y,Xs) ).

% list_update_code(2)
tff(fact_4458_set__encode__eq,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( ( aa_set_nat_nat(nat_set_encode,A4) = aa_set_nat_nat(nat_set_encode,B5) )
        <=> ( A4 = B5 ) ) ) ) ).

% set_encode_eq
tff(fact_4459_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_4460_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = some_nat(the_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_4461_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = some_num(the_num(Option)) ) ) ).

% option.exhaust_sel
tff(fact_4462_eucl__rel__int__dividesI,axiom,
    ! [L: int,K2: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K2 = aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L) )
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K2,L),product_Pair_int_int(Q2,zero_zero_int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_4463_set__update__memI,axiom,
    ! [N: nat,Xs: list_complex,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3451745648224563538omplex(Xs)))
     => pp(aa_set_complex_bool(member_complex(X),set_complex2(list_update_complex(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_4464_set__update__memI,axiom,
    ! [N: nat,Xs: list_real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => pp(aa_set_real_bool(member_real(X),set_real2(list_update_real(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_4465_set__update__memI,axiom,
    ! [N: nat,Xs: list_set_nat,X: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
     => member_set_nat(X,set_set_nat2(list_update_set_nat(Xs,N,X))) ) ).

% set_update_memI
tff(fact_4466_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_4467_set__update__memI,axiom,
    ! [N: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => member_bool(X,set_bool2(list_update_bool(Xs,N,X))) ) ).

% set_update_memI
tff(fact_4468_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
     => pp(aa_set_nat_bool(member_nat(X),set_nat2(list_update_nat(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_4469_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => pp(aa_set_int_bool(member_int(X),set_int2(list_update_int(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_4470_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I2,X) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I2) = X ) ) ) ).

% list_update_same_conv
tff(fact_4471_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
     => ( ( list_update_bool(Xs,I2,X) = Xs )
      <=> ( pp(aa_nat_bool(nth_bool(Xs),I2))
        <=> pp(X) ) ) ) ).

% list_update_same_conv
tff(fact_4472_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( ( list_update_nat(Xs,I2,X) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I2) = X ) ) ) ).

% list_update_same_conv
tff(fact_4473_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
     => ( ( list_update_int(Xs,I2,X) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I2) = X ) ) ) ).

% list_update_same_conv
tff(fact_4474_nth__list__update,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,J2: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( ( I2 = J2 )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,X),J2) = X ) )
        & ( ( I2 != J2 )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,X),J2) = nth_VEBT_VEBT(Xs,J2) ) ) ) ) ).

% nth_list_update
tff(fact_4475_nth__list__update,axiom,
    ! [I2: nat,Xs: list_bool,X: bool,J2: 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(list_update_bool(Xs,I2,X)),J2))
      <=> ( ( ( I2 = J2 )
           => pp(X) )
          & ( ( I2 != J2 )
           => pp(aa_nat_bool(nth_bool(Xs),J2)) ) ) ) ) ).

% nth_list_update
tff(fact_4476_nth__list__update,axiom,
    ! [I2: nat,Xs: list_nat,J2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( ( ( I2 = J2 )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I2,X)),J2) = X ) )
        & ( ( I2 != J2 )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I2,X)),J2) = aa_nat_nat(nth_nat(Xs),J2) ) ) ) ) ).

% nth_list_update
tff(fact_4477_nth__list__update,axiom,
    ! [I2: nat,Xs: list_int,J2: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
     => ( ( ( I2 = J2 )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I2,X)),J2) = X ) )
        & ( ( I2 != J2 )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I2,X)),J2) = aa_nat_int(nth_int(Xs),J2) ) ) ) ) ).

% nth_list_update
tff(fact_4478_set__encode__inf,axiom,
    ! [A4: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( aa_set_nat_nat(nat_set_encode,A4) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_4479_eucl__rel__int__iff,axiom,
    ! [K2: int,L: int,Q2: int,R: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K2,L),product_Pair_int_int(Q2,R)))
    <=> ( ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q2)),R) )
        & ( 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),R))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),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),R))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R),zero_zero_int)) ) )
            & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( Q2 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_4480_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),product_Pair_int_int(Q2,R)))
       => pp(aa_Pro4501881432599481050t_bool(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(Q2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R))))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_4481_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_4482_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_4483_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,aa_option_nat_bool(aa_opt7943525813896274546t_bool(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))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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_4484_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,aa_option_nat_bool(aa_opt7943525813896274546t_bool(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))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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_4485_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,aa_option_nat_bool(aa_opt7943525813896274546t_bool(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))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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_4486_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,aa_option_nat_bool(aa_opt7943525813896274546t_bool(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))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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_4487_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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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,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_n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t2,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)))))),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_4488_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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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_4489_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(vEBT_vebt_member(T)) ).

% set_vebt'_def
tff(fact_4490_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_pred(T,X) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_af(T),X)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_4491_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
     => ( ( vEBT_vebt_succ(T,X) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ag(T),X)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_4492_max_Obounded__iff,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C2)),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,C2),A)) ) ) ).

% max.bounded_iff
tff(fact_4493_max_Obounded__iff,axiom,
    ! [B: num,C2: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C2)),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,C2),A)) ) ) ).

% max.bounded_iff
tff(fact_4494_max_Obounded__iff,axiom,
    ! [B: nat,C2: 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),C2)),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,C2),A)) ) ) ).

% max.bounded_iff
tff(fact_4495_max_Obounded__iff,axiom,
    ! [B: int,C2: 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),C2)),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,C2),A)) ) ) ).

% max.bounded_iff
tff(fact_4496_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(aa_rat_fun_rat_rat(ord_max_rat,A),B) = B ) ) ).

% max.absorb2
tff(fact_4497_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(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb2
tff(fact_4498_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_4499_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_4500_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(aa_rat_fun_rat_rat(ord_max_rat,A),B) = A ) ) ).

% max.absorb1
tff(fact_4501_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(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb1
tff(fact_4502_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_4503_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_4504_max__bot,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,bot_bot_set_nat),X) = X ).

% max_bot
tff(fact_4505_max__bot,axiom,
    ! [X: set_int] : ord_max_set_int(bot_bot_set_int,X) = X ).

% max_bot
tff(fact_4506_max__bot,axiom,
    ! [X: set_real] : ord_max_set_real(bot_bot_set_real,X) = X ).

% max_bot
tff(fact_4507_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_4508_max__bot2,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,X),bot_bot_set_nat) = X ).

% max_bot2
tff(fact_4509_max__bot2,axiom,
    ! [X: set_int] : ord_max_set_int(X,bot_bot_set_int) = X ).

% max_bot2
tff(fact_4510_max__bot2,axiom,
    ! [X: set_real] : ord_max_set_real(X,bot_bot_set_real) = X ).

% max_bot2
tff(fact_4511_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_4512_max__Suc__Suc,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M2),N)) ).

% max_Suc_Suc
tff(fact_4513_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_4514_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_4515_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_4516_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_4517_max__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N) = N ).

% max_0L
tff(fact_4518_max__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),zero_zero_nat) = N ).

% max_0R
tff(fact_4519_finite__Collect__le__nat,axiom,
    ! [K2: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ah,K2)))) ).

% finite_Collect_le_nat
tff(fact_4520_max__number__of_I1_J,axiom,
    ! [U: num,V2: num] :
      ( ( ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V2))
       => ( ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V2)) = numera1916890842035813515d_enat(V2) ) )
      & ( ~ ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V2))
       => ( ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V2)) = numera1916890842035813515d_enat(U) ) ) ) ).

% max_number_of(1)
tff(fact_4521_max__number__of_I1_J,axiom,
    ! [U: num,V2: 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,V2)))
       => ( aa_real_real(ord_max_real(aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V2)) = aa_num_real(numeral_numeral_real,V2) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V2)))
       => ( aa_real_real(ord_max_real(aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V2)) = aa_num_real(numeral_numeral_real,U) ) ) ) ).

% max_number_of(1)
tff(fact_4522_max__number__of_I1_J,axiom,
    ! [U: num,V2: 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,V2)))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V2)) = aa_num_rat(numeral_numeral_rat,V2) ) )
      & ( ~ 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,V2)))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V2)) = aa_num_rat(numeral_numeral_rat,U) ) ) ) ).

% max_number_of(1)
tff(fact_4523_max__number__of_I1_J,axiom,
    ! [U: num,V2: 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,V2)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V2)) = aa_num_nat(numeral_numeral_nat,V2) ) )
      & ( ~ 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,V2)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V2)) = aa_num_nat(numeral_numeral_nat,U) ) ) ) ).

% max_number_of(1)
tff(fact_4524_max__number__of_I1_J,axiom,
    ! [U: num,V2: 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,V2)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V2)) = aa_num_int(numeral_numeral_int,V2) ) )
      & ( ~ 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,V2)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V2)) = aa_num_int(numeral_numeral_int,U) ) ) ) ).

% max_number_of(1)
tff(fact_4525_max__0__1_I3_J,axiom,
    ! [X: num] : aa_rat_rat(aa_rat_fun_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_4526_max__0__1_I3_J,axiom,
    ! [X: num] : ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).

% max_0_1(3)
tff(fact_4527_max__0__1_I3_J,axiom,
    ! [X: num] : aa_real_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_4528_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_4529_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_4530_max__0__1_I4_J,axiom,
    ! [X: num] : aa_rat_rat(aa_rat_fun_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_4531_max__0__1_I4_J,axiom,
    ! [X: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(X),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(X) ).

% max_0_1(4)
tff(fact_4532_max__0__1_I4_J,axiom,
    ! [X: num] : aa_real_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_4533_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_4534_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_4535_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_4536_max__0__1_I2_J,axiom,
    aa_real_real(ord_max_real(one_one_real),zero_zero_real) = one_one_real ).

% max_0_1(2)
tff(fact_4537_max__0__1_I2_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,one_one_rat),zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_4538_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_4539_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_4540_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_4541_max__0__1_I1_J,axiom,
    aa_real_real(ord_max_real(zero_zero_real),one_one_real) = one_one_real ).

% max_0_1(1)
tff(fact_4542_max__0__1_I1_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,zero_zero_rat),one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_4543_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_4544_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_4545_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_4546_max__0__1_I5_J,axiom,
    ! [X: num] : ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).

% max_0_1(5)
tff(fact_4547_max__0__1_I5_J,axiom,
    ! [X: num] : aa_real_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_4548_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_4549_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_4550_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_4551_max__0__1_I6_J,axiom,
    ! [X: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(X),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(X) ).

% max_0_1(6)
tff(fact_4552_max__0__1_I6_J,axiom,
    ! [X: num] : aa_real_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_4553_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_4554_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_4555_card__Collect__le__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ah,N))) = aa_nat_nat(suc,N) ).

% card_Collect_le_nat
tff(fact_4556_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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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_4557_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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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_4558_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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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_4559_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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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_4560_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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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_4561_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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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_4562_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(ord_max_real(aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).

% of_nat_max
tff(fact_4563_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_4564_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_4565_max__def__raw,axiom,
    ! [X5: set_nat,Xa: set_nat] :
      ( ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X5),Xa))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,X5),Xa) = Xa ) )
      & ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(X5),Xa))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,X5),Xa) = X5 ) ) ) ).

% max_def_raw
tff(fact_4566_max__def__raw,axiom,
    ! [X5: rat,Xa: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Xa))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X5),Xa) = Xa ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Xa))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X5),Xa) = X5 ) ) ) ).

% max_def_raw
tff(fact_4567_max__def__raw,axiom,
    ! [X5: num,Xa: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Xa))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X5),Xa) = Xa ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Xa))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X5),Xa) = X5 ) ) ) ).

% max_def_raw
tff(fact_4568_max__def__raw,axiom,
    ! [X5: nat,Xa: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Xa))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X5),Xa) = Xa ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Xa))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X5),Xa) = X5 ) ) ) ).

% max_def_raw
tff(fact_4569_max__def__raw,axiom,
    ! [X5: int,Xa: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),Xa))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X5),Xa) = Xa ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),Xa))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X5),Xa) = X5 ) ) ) ).

% max_def_raw
tff(fact_4570_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_4571_lambda__zero,axiom,
    aTP_Lamm_ai = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_4572_lambda__zero,axiom,
    aTP_Lamm_aj = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_4573_lambda__zero,axiom,
    aTP_Lamm_ak = aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat) ).

% lambda_zero
tff(fact_4574_lambda__zero,axiom,
    aTP_Lamm_al = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_4575_lambda__zero,axiom,
    aTP_Lamm_am = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_4576_lambda__one,axiom,
    aTP_Lamm_an = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer) ).

% lambda_one
tff(fact_4577_lambda__one,axiom,
    aTP_Lamm_ao = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_4578_lambda__one,axiom,
    aTP_Lamm_ap = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_4579_lambda__one,axiom,
    aTP_Lamm_aq = aa_rat_fun_rat_rat(times_times_rat,one_one_rat) ).

% lambda_one
tff(fact_4580_lambda__one,axiom,
    aTP_Lamm_ar = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_4581_lambda__one,axiom,
    aTP_Lamm_as = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_4582_finite__M__bounded__by__nat,axiom,
    ! [P2: fun_nat_bool,I2: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_at(P2),I2)))) ).

% finite_M_bounded_by_nat
tff(fact_4583_finite__less__ub,axiom,
    ! [F2: fun_nat_nat,U: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N3),aa_nat_nat(F2,N3)))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_au(F2),U)))) ) ).

% finite_less_ub
tff(fact_4584_max_Omono,axiom,
    ! [C2: rat,A: rat,D: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),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(aa_rat_fun_rat_rat(ord_max_rat,C2),D)),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B))) ) ) ).

% max.mono
tff(fact_4585_max_Omono,axiom,
    ! [C2: num,A: num,D: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),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(aa_num_fun_num_num(ord_max_num,C2),D)),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ) ).

% max.mono
tff(fact_4586_max_Omono,axiom,
    ! [C2: nat,A: nat,D: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),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,C2),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ) ).

% max.mono
tff(fact_4587_max_Omono,axiom,
    ! [C2: int,A: int,D: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),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,C2),D)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ) ).

% max.mono
tff(fact_4588_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(aa_rat_fun_rat_rat(ord_max_rat,A),B) ) ) ).

% max.orderE
tff(fact_4589_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(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.orderE
tff(fact_4590_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_4591_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_4592_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_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_4593_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A = aa_num_num(aa_num_fun_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_4594_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_4595_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_4596_max_OboundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C2)),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,C2),A)) ) ) ).

% max.boundedE
tff(fact_4597_max_OboundedE,axiom,
    ! [B: num,C2: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C2)),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,C2),A)) ) ) ).

% max.boundedE
tff(fact_4598_max_OboundedE,axiom,
    ! [B: nat,C2: 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),C2)),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,C2),A)) ) ) ).

% max.boundedE
tff(fact_4599_max_OboundedE,axiom,
    ! [B: int,C2: 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),C2)),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,C2),A)) ) ) ).

% max.boundedE
tff(fact_4600_max_OboundedI,axiom,
    ! [B: rat,A: rat,C2: 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,C2),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C2)),A)) ) ) ).

% max.boundedI
tff(fact_4601_max_OboundedI,axiom,
    ! [B: num,A: num,C2: 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,C2),A))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C2)),A)) ) ) ).

% max.boundedI
tff(fact_4602_max_OboundedI,axiom,
    ! [B: nat,A: nat,C2: 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,C2),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),C2)),A)) ) ) ).

% max.boundedI
tff(fact_4603_max_OboundedI,axiom,
    ! [B: int,A: int,C2: 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,C2),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),C2)),A)) ) ) ).

% max.boundedI
tff(fact_4604_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(aa_rat_fun_rat_rat(ord_max_rat,A),B) ) ) ).

% max.order_iff
tff(fact_4605_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(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.order_iff
tff(fact_4606_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_4607_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_4608_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B))) ).

% max.cobounded1
tff(fact_4609_max_Ocobounded1,axiom,
    ! [A: num,B: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ).

% max.cobounded1
tff(fact_4610_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_4611_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_4612_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B))) ).

% max.cobounded2
tff(fact_4613_max_Ocobounded2,axiom,
    ! [B: num,A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ).

% max.cobounded2
tff(fact_4614_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_4615_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_4616_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(aa_rat_fun_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_4617_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(aa_num_fun_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_4618_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_4619_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_4620_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(aa_rat_fun_rat_rat(ord_max_rat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_4621_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(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_4622_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_4623_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_4624_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(aa_rat_fun_rat_rat(ord_max_rat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_4625_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(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_4626_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_4627_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_4628_max_OcoboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B))) ) ).

% max.coboundedI1
tff(fact_4629_max_OcoboundedI1,axiom,
    ! [C2: num,A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.coboundedI1
tff(fact_4630_max_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.coboundedI1
tff(fact_4631_max_OcoboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.coboundedI1
tff(fact_4632_max_OcoboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B))) ) ).

% max.coboundedI2
tff(fact_4633_max_OcoboundedI2,axiom,
    ! [C2: num,B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C2),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.coboundedI2
tff(fact_4634_max_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.coboundedI2
tff(fact_4635_max_OcoboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.coboundedI2
tff(fact_4636_max__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_4637_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(aa_rat_fun_rat_rat(ord_max_rat,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_4638_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(aa_num_fun_num_num(ord_max_num,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_4639_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_4640_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_4641_max__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,X),Y) = X ) ) ).

% max_absorb1
tff(fact_4642_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(aa_rat_fun_rat_rat(ord_max_rat,X),Y) = X ) ) ).

% max_absorb1
tff(fact_4643_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(aa_num_fun_num_num(ord_max_num,X),Y) = X ) ) ).

% max_absorb1
tff(fact_4644_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_4645_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_4646_max__def,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,A),B) = B ) )
      & ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,A),B) = A ) ) ) ).

% max_def
tff(fact_4647_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(aa_rat_fun_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(aa_rat_fun_rat_rat(ord_max_rat,A),B) = A ) ) ) ).

% max_def
tff(fact_4648_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(aa_num_fun_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(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ) ).

% max_def
tff(fact_4649_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_4650_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_4651_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z2: real] : aa_real_real(plus_plus_real(aa_real_real(ord_max_real(X),Y)),Z2) = aa_real_real(ord_max_real(aa_real_real(plus_plus_real(X),Z2)),aa_real_real(plus_plus_real(Y),Z2)) ).

% max_add_distrib_left
tff(fact_4652_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y)),Z2) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Z2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),Z2)) ).

% max_add_distrib_left
tff(fact_4653_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_4654_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_4655_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z2: real] : aa_real_real(plus_plus_real(X),aa_real_real(ord_max_real(Y),Z2)) = aa_real_real(ord_max_real(aa_real_real(plus_plus_real(X),Y)),aa_real_real(plus_plus_real(X),Z2)) ).

% max_add_distrib_right
tff(fact_4656_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,Y),Z2)) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Z2)) ).

% max_add_distrib_right
tff(fact_4657_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_4658_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_4659_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y)),Z2) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Z2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Y),Z2)) ).

% max_diff_distrib_left
tff(fact_4660_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_4661_nat__add__max__left,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M2),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Q2)) ).

% nat_add_max_left
tff(fact_4662_nat__add__max__right,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Q2)) ).

% nat_add_max_right
tff(fact_4663_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T)) ).

% set_vebt_def
tff(fact_4664_nat__mult__max__left,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M2),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).

% nat_mult_max_left
tff(fact_4665_nat__mult__max__right,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Q2)) ).

% nat_mult_max_right
tff(fact_4666_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N)) ).

% numeral_code(2)
tff(fact_4667_numeral__code_I2_J,axiom,
    ! [N: num] : numera1916890842035813515d_enat(aa_num_num(bit0,N)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N),numera1916890842035813515d_enat(N)) ).

% numeral_code(2)
tff(fact_4668_numeral__code_I2_J,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit0,N)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N)) ).

% numeral_code(2)
tff(fact_4669_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,N)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N)) ).

% numeral_code(2)
tff(fact_4670_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_code(2)
tff(fact_4671_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).

% numeral_code(2)
tff(fact_4672_nat__leq__as__int,axiom,
    ! [X5: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_leq_as_int
tff(fact_4673_power__numeral__even,axiom,
    ! [Z2: complex,W2: num] : aa_nat_complex(power_power_complex(Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W2))) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Z2),aa_num_nat(numeral_numeral_nat,W2))),aa_nat_complex(power_power_complex(Z2),aa_num_nat(numeral_numeral_nat,W2))) ).

% power_numeral_even
tff(fact_4674_power__numeral__even,axiom,
    ! [Z2: real,W2: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),aa_num_nat(numeral_numeral_nat,W2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),aa_num_nat(numeral_numeral_nat,W2))) ).

% power_numeral_even
tff(fact_4675_power__numeral__even,axiom,
    ! [Z2: rat,W2: num] : aa_nat_rat(power_power_rat(Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Z2),aa_num_nat(numeral_numeral_nat,W2))),aa_nat_rat(power_power_rat(Z2),aa_num_nat(numeral_numeral_nat,W2))) ).

% power_numeral_even
tff(fact_4676_power__numeral__even,axiom,
    ! [Z2: nat,W2: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z2),aa_num_nat(numeral_numeral_nat,W2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z2),aa_num_nat(numeral_numeral_nat,W2))) ).

% power_numeral_even
tff(fact_4677_power__numeral__even,axiom,
    ! [Z2: int,W2: num] : aa_nat_int(power_power_int(Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Z2),aa_num_nat(numeral_numeral_nat,W2))),aa_nat_int(power_power_int(Z2),aa_num_nat(numeral_numeral_nat,W2))) ).

% power_numeral_even
tff(fact_4678_card__less,axiom,
    ! [M5: set_nat,I2: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M5))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_av(M5),I2))) != zero_zero_nat ) ) ).

% card_less
tff(fact_4679_card__less__Suc,axiom,
    ! [M5: set_nat,I2: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M5))
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_aw(M5),I2)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_av(M5),I2))) ) ) ).

% card_less_Suc
tff(fact_4680_card__less__Suc2,axiom,
    ! [M5: set_nat,I2: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M5))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_aw(M5),I2))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_av(M5),I2))) ) ) ).

% card_less_Suc2
tff(fact_4681_nat__minus__add__max,axiom,
    ! [N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)),M2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),M2) ).

% nat_minus_add_max
tff(fact_4682_finite__lists__length__eq,axiom,
    ! [A4: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_ax(A4),N))) ) ).

% finite_lists_length_eq
tff(fact_4683_finite__lists__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A4)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_ay(A4),N))) ) ).

% finite_lists_length_eq
tff(fact_4684_finite__lists__length__eq,axiom,
    ! [A4: set_bool,N: nat] :
      ( finite_finite_bool(A4)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_az(A4),N))) ) ).

% finite_lists_length_eq
tff(fact_4685_finite__lists__length__eq,axiom,
    ! [A4: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_ba(A4),N))) ) ).

% finite_lists_length_eq
tff(fact_4686_finite__lists__length__eq,axiom,
    ! [A4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_bb(A4),N))) ) ).

% finite_lists_length_eq
tff(fact_4687_card__lists__length__eq,axiom,
    ! [A4: set_list_nat,N: nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( finite7325466520557071688st_nat(collec5989764272469232197st_nat(aa_nat3401988846471794802t_bool(aTP_Lamm_bc(A4),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_list_nat(A4)),N) ) ) ).

% card_lists_length_eq
tff(fact_4688_card__lists__length__eq,axiom,
    ! [A4: set_set_nat,N: nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( finite5631907774883551598et_nat(collect_list_set_nat(aa_nat948039967059030092t_bool(aTP_Lamm_bd(A4),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_set_nat(A4)),N) ) ) ).

% card_lists_length_eq
tff(fact_4689_card__lists__length__eq,axiom,
    ! [A4: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( finite5120063068150530198omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_ax(A4),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_complex(A4)),N) ) ) ).

% card_lists_length_eq
tff(fact_4690_card__lists__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A4)
     => ( finite5915292604075114978T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_ay(A4),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite7802652506058667612T_VEBT(A4)),N) ) ) ).

% card_lists_length_eq
tff(fact_4691_card__lists__length__eq,axiom,
    ! [A4: set_bool,N: nat] :
      ( finite_finite_bool(A4)
     => ( finite1699927739564962373t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_az(A4),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_bool(A4)),N) ) ) ).

% card_lists_length_eq
tff(fact_4692_card__lists__length__eq,axiom,
    ! [A4: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( finite_card_list_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_ba(A4),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_int(A4)),N) ) ) ).

% card_lists_length_eq
tff(fact_4693_card__lists__length__eq,axiom,
    ! [A4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_bb(A4),N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set_nat_nat(finite_card_nat,A4)),N) ) ) ).

% card_lists_length_eq
tff(fact_4694_finite__lists__length__le,axiom,
    ! [A4: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_be(A4),N))) ) ).

% finite_lists_length_le
tff(fact_4695_finite__lists__length__le,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A4)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_bf(A4),N))) ) ).

% finite_lists_length_le
tff(fact_4696_finite__lists__length__le,axiom,
    ! [A4: set_bool,N: nat] :
      ( finite_finite_bool(A4)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_bg(A4),N))) ) ).

% finite_lists_length_le
tff(fact_4697_finite__lists__length__le,axiom,
    ! [A4: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_bh(A4),N))) ) ).

% finite_lists_length_le
tff(fact_4698_finite__lists__length__le,axiom,
    ! [A4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_bi(A4),N))) ) ).

% finite_lists_length_le
tff(fact_4699_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va3: 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,Va3)),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,Va3))),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,Va3)),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3)),TreeList2,Summary)) ).

% vebt_insert.simps(5)
tff(fact_4700_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S2)),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,V2)),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,V2)),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,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,X),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(TreeList2))) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_4701_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2)),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,V2)),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,V2)),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,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,X),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(TreeList2))) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_4702_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,S: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info,zero_zero_nat,Ts,S) )
             => ( Y != vEBT_Node(Info,zero_zero_nat,Ts,S) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S) )
               => ( Y != vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S) ) )
           => ( ! [V: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),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,V)),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_4703_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va3: 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,Va3)),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,Va3))),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,Va3))),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,Va3))),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_4704_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V2: 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,V2),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,V2)),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,V2)),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,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,X),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(TreeList2))) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_4705_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,Uw: vEBT_VEBT] : X != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw)
         => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : X = vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,S)
               => ( ( 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,V)),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,V)),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,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,Xa2),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(TreeList))) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_4706_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 ) ) ) ) )
       => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S: vEBT_VEBT] : X = vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,S)
             => ~ ( ( 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,V)),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,V)),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,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,Xa2),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(TreeList))) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_4707_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,Uw: vEBT_VEBT] : X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw)
           => pp(Y) )
         => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : X = vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,S)
               => ( 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,V)),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,V)),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,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,Xa2),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(TreeList))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_4708_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] :
            ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2)
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V: 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,V),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,V)),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,V)),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,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,Xa2),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(TreeList))) ) ) )
         => ~ ! [V: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd)
               => ~ ( ( 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,V)),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,V)),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,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,Xa2),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(TreeList))) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_4709_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_4710_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)
       => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2)
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V: 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,V),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,V)),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,V)),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,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,Xa2),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(TreeList))) ) ) )
             => ~ ! [V: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd)
                   => ( ( 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,V)),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,V)),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,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,Xa2),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(TreeList))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_4711_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) )
       => ( ( ? [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)
           => pp(Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2)
               => ( pp(Y)
                <=> ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V: 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,V),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,V)),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,V)),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,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,Xa2),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(TreeList))) ) ) ) )
             => ~ ! [V: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd)
                   => ( 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,V)),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,V)),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,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,Xa2),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(TreeList))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_4712_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,Uw: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw)
         => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz2)
           => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V),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_4713_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,Uw: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw)
           => pp(Y) )
         => ( ( ? [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz2)
             => pp(Y) )
           => ( ( ? [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V),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_4714_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va3: 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,Va3)),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,Va3)),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,Va3))),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,aa_option_nat_bool(aa_opt7943525813896274546t_bool(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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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,Va3))),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,Va3))),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,Va3))),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_4715_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va3: 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,Va3)),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,Va3)),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,Va3))),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,aa_option_nat_bool(aa_opt7943525813896274546t_bool(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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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,Va3))),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,Va3))),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,Va3))),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_4716_vebt__delete_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va3: 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,Va3)),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,Va3)),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,Va3)),TreeList2,Summary),X) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),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,Va3)),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3))),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,Va3)),TreeList2,Summary)) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_4717_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) )
               => ( ? [N3: nat] : Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,N3))
                 => ( 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,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst,Smry) )
                   => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst,Smry) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm) )
                     => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm) ) )
                 => ~ ! [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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) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_4718_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,Uw: bool] : X = vEBT_Leaf(Uv2,Uw)
           => ( ? [N3: nat] : Xa2 = aa_nat_nat(suc,N3)
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy,Uz2)
             => ( Y != none_nat ) )
           => ( ( ? [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2,Vd)
               => ( Y != none_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg,Vh)
                 => ( 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,aa_option_nat_bool(aa_opt7943525813896274546t_bool(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))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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_4719_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] :
              ( ? [Uw: bool] : X = vEBT_Leaf(A3,Uw)
             => ( ( 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 ) ) ) ) ) ) )
           => ( ( ? [Uy: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz2,Va2)
               => ( Y != none_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve)
                 => ( Y != none_nat ) )
               => ( ( ? [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh,Vi)
                   => ( 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,aa_option_nat_bool(aa_opt7943525813896274546t_bool(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))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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_4720_finite__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_bj(N),C2)))) ) ).

% finite_nth_roots
tff(fact_4721_card__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(collect_real(aa_nat_fun_real_bool(aTP_Lamm_bk,N)))),N)) ) ).

% card_roots_unity
tff(fact_4722_card__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(collect_complex(aTP_Lamm_bl(N)))),N)) ) ).

% card_roots_unity
tff(fact_4723_finite__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_bk,N)))) ) ).

% finite_roots_unity
tff(fact_4724_finite__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aTP_Lamm_bl(N)))) ) ).

% finite_roots_unity
tff(fact_4725_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,Uw: bool] :
                ( ( X = vEBT_Leaf(Uv2,Uw) )
               => ! [N3: nat] :
                    ( ( Xa2 = aa_nat_nat(suc,N3) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uv2,Uw),aa_nat_nat(suc,N3))) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy,Uz2) )
                 => ( ( Y = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy,Uz2),Xa2)) ) )
             => ( ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2,Vd) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2,Vd),Xa2)) ) )
               => ( ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg,Vh) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),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,aa_option_nat_bool(aa_opt7943525813896274546t_bool(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))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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_4726_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,Uw: bool] :
                ( ( X = vEBT_Leaf(A3,Uw) )
               => ( ( 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,Uw),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)))) ) ) )
             => ( ! [Uy: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz2,Va2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz2,Va2),Xa2)) ) )
               => ( ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve),Xa2)) ) )
                 => ( ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh,Vi) )
                       => ( ( Y = none_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),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,aa_option_nat_bool(aa_opt7943525813896274546t_bool(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))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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_4727_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) )
                 => ! [N3: nat] :
                      ( ( Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
                     => ( ( 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,N3)))) ) ) )
             => ( ! [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,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst,Smry) )
                     => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst,Smry) )
                       => ~ 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,TrLst,Smry),Xa2)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm) )
                       => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm) )
                         => ~ 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),Tr,Sm),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 = 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(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)))))),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_4728_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,S: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,zero_zero_nat,Ts,S) )
               => ( ( Y = vEBT_Node(Info,zero_zero_nat,Ts,S) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info,zero_zero_nat,Ts,S),Xa2)) ) )
           => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S) )
                 => ( ( Y = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S),Xa2)) ) )
             => ( ! [V: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),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,V)),TreeList,Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),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_4729_low__def,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N) = 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))),N)) ).

% low_def
tff(fact_4730_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_4731_mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_4732_mod__0,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(zero_z2226904508553997617atural,A) = zero_z2226904508553997617atural ).

% mod_0
tff(fact_4733_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_4734_mod__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),zero_zero_int) = A ).

% mod_by_0
tff(fact_4735_mod__by__0,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(A,zero_z2226904508553997617atural) = A ).

% mod_by_0
tff(fact_4736_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_4737_mod__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),A) = zero_zero_int ).

% mod_self
tff(fact_4738_mod__self,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(A,A) = zero_z2226904508553997617atural ).

% mod_self
tff(fact_4739_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_4740_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_4741_bits__mod__0,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(zero_z2226904508553997617atural,A) = zero_z2226904508553997617atural ).

% bits_mod_0
tff(fact_4742_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_4743_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_4744_mod__mult__self2__is__0,axiom,
    ! [A: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),B) = zero_z2226904508553997617atural ).

% mod_mult_self2_is_0
tff(fact_4745_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_4746_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),B) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_4747_mod__mult__self1__is__0,axiom,
    ! [B: code_natural,A: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A),B) = zero_z2226904508553997617atural ).

% mod_mult_self1_is_0
tff(fact_4748_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_4749_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_4750_mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_4751_mod__by__1,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(A,one_one_Code_natural) = zero_z2226904508553997617atural ).

% mod_by_1
tff(fact_4752_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_4753_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_4754_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_4755_bits__mod__by__1,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(A,one_one_Code_natural) = zero_z2226904508553997617atural ).

% bits_mod_by_1
tff(fact_4756_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_4757_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_4758_mod__div__trivial,axiom,
    ! [A: code_natural,B: code_natural] : divide5121882707175180666atural(modulo8411746178871703098atural(A,B),B) = zero_z2226904508553997617atural ).

% mod_div_trivial
tff(fact_4759_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_4760_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_4761_bits__mod__div__trivial,axiom,
    ! [A: code_natural,B: code_natural] : divide5121882707175180666atural(modulo8411746178871703098atural(A,B),B) = zero_z2226904508553997617atural ).

% bits_mod_div_trivial
tff(fact_4762_mod__mult__self1,axiom,
    ! [A: nat,C2: 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,C2),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_4763_mod__mult__self1,axiom,
    ! [A: int,C2: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self1
tff(fact_4764_mod__mult__self1,axiom,
    ! [A: code_natural,C2: code_natural,B: code_natural] : modulo8411746178871703098atural(plus_p4538020629002901425atural(A,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C2),B)),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self1
tff(fact_4765_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C2: 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),C2))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_4766_mod__mult__self2,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self2
tff(fact_4767_mod__mult__self2,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] : modulo8411746178871703098atural(plus_p4538020629002901425atural(A,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C2)),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self2
tff(fact_4768_mod__mult__self3,axiom,
    ! [C2: 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,C2),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_4769_mod__mult__self3,axiom,
    ! [C2: int,B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self3
tff(fact_4770_mod__mult__self3,axiom,
    ! [C2: code_natural,B: code_natural,A: code_natural] : modulo8411746178871703098atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C2),B),A),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self3
tff(fact_4771_mod__mult__self4,axiom,
    ! [B: nat,C2: 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),C2)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_4772_mod__mult__self4,axiom,
    ! [B: int,C2: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self4
tff(fact_4773_mod__mult__self4,axiom,
    ! [B: code_natural,C2: code_natural,A: code_natural] : modulo8411746178871703098atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C2),A),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self4
tff(fact_4774_mod__by__Suc__0,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_4775_Suc__mod__mult__self4,axiom,
    ! [N: nat,K2: nat,M2: 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,N),K2)),M2))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M2)),N) ).

% Suc_mod_mult_self4
tff(fact_4776_Suc__mod__mult__self3,axiom,
    ! [K2: nat,N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)),M2))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M2)),N) ).

% Suc_mod_mult_self3
tff(fact_4777_Suc__mod__mult__self2,axiom,
    ! [M2: nat,N: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K2)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M2)),N) ).

% Suc_mod_mult_self2
tff(fact_4778_Suc__mod__mult__self1,axiom,
    ! [M2: nat,K2: nat,N: 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,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M2)),N) ).

% Suc_mod_mult_self1
tff(fact_4779_mod2__Suc__Suc,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_4780_Suc__times__numeral__mod__eq,axiom,
    ! [K2: num,N: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K2) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K2)),N))),aa_num_nat(numeral_numeral_nat,K2)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_4781_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_4782_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_4783_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4784_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) != zero_z2226904508553997617atural )
    <=> ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = one_one_Code_natural ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4785_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_4786_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_4787_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4788_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) != one_one_Code_natural )
    <=> ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = zero_z2226904508553997617atural ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4789_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_4790_add__self__mod__2,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),M2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_4791_mod2__gr__0,axiom,
    ! [M2: 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,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_4792_mod__mult__eq,axiom,
    ! [A: nat,C2: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) ).

% mod_mult_eq
tff(fact_4793_mod__mult__eq,axiom,
    ! [A: int,C2: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C2))),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2) ).

% mod_mult_eq
tff(fact_4794_mod__mult__eq,axiom,
    ! [A: code_natural,C2: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,modulo8411746178871703098atural(A,C2)),modulo8411746178871703098atural(B,C2)),C2) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),C2) ).

% mod_mult_eq
tff(fact_4795_mod__mult__cong,axiom,
    ! [A: nat,C2: nat,A2: nat,B: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) ) ) ) ).

% mod_mult_cong
tff(fact_4796_mod__mult__cong,axiom,
    ! [A: int,C2: int,A2: int,B: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C2) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C2) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C2) ) ) ) ).

% mod_mult_cong
tff(fact_4797_mod__mult__cong,axiom,
    ! [A: code_natural,C2: code_natural,A2: code_natural,B: code_natural,B2: code_natural] :
      ( ( modulo8411746178871703098atural(A,C2) = modulo8411746178871703098atural(A2,C2) )
     => ( ( modulo8411746178871703098atural(B,C2) = modulo8411746178871703098atural(B2,C2) )
       => ( modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),C2) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A2),B2),C2) ) ) ) ).

% mod_mult_cong
tff(fact_4798_mod__mult__mult2,axiom,
    ! [A: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C2) ).

% mod_mult_mult2
tff(fact_4799_mod__mult__mult2,axiom,
    ! [A: int,C2: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),C2) ).

% mod_mult_mult2
tff(fact_4800_mod__mult__mult2,axiom,
    ! [A: code_natural,C2: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C2),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C2)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,modulo8411746178871703098atural(A,B)),C2) ).

% mod_mult_mult2
tff(fact_4801_mult__mod__right,axiom,
    ! [C2: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ).

% mult_mod_right
tff(fact_4802_mult__mod__right,axiom,
    ! [C2: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,C2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ).

% mult_mod_right
tff(fact_4803_mult__mod__right,axiom,
    ! [C2: code_natural,A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C2),modulo8411746178871703098atural(A,B)) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C2),A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C2),B)) ).

% mult_mod_right
tff(fact_4804_mod__mult__left__eq,axiom,
    ! [A: nat,C2: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C2)),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) ).

% mod_mult_left_eq
tff(fact_4805_mod__mult__left__eq,axiom,
    ! [A: int,C2: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C2)),B)),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2) ).

% mod_mult_left_eq
tff(fact_4806_mod__mult__left__eq,axiom,
    ! [A: code_natural,C2: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,modulo8411746178871703098atural(A,C2)),B),C2) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),C2) ).

% mod_mult_left_eq
tff(fact_4807_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) ).

% mod_mult_right_eq
tff(fact_4808_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C2))),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2) ).

% mod_mult_right_eq
tff(fact_4809_mod__mult__right__eq,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),modulo8411746178871703098atural(B,C2)),C2) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),C2) ).

% mod_mult_right_eq
tff(fact_4810_mod__Suc__Suc__eq,axiom,
    ! [M2: nat,N: 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,M2),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M2))),N) ).

% mod_Suc_Suc_eq
tff(fact_4811_mod__Suc__eq,axiom,
    ! [M2: nat,N: 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,M2),N))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M2)),N) ).

% mod_Suc_eq
tff(fact_4812_mod__less__eq__dividend,axiom,
    ! [M2: nat,N: 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,M2),N)),M2)) ).

% mod_less_eq_dividend
tff(fact_4813_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_4814_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(aa_int_fun_int_int(modulo_modulo_int,A),B)),A)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4815_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_4816_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(aa_int_fun_int_int(modulo_modulo_int,A),B)),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4817_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_4818_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4819_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural(A,B) = A )
    <=> ( divide5121882707175180666atural(A,B) = zero_z2226904508553997617atural ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4820_cong__exp__iff__simps_I9_J,axiom,
    ! [M2: num,Q2: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_4821_cong__exp__iff__simps_I9_J,axiom,
    ! [M2: num,Q2: num,N: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_4822_cong__exp__iff__simps_I4_J,axiom,
    ! [M2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_4823_cong__exp__iff__simps_I4_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_4824_mod__eqE,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C2) )
     => ~ ! [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,C2),D5)) ) ).

% mod_eqE
tff(fact_4825_mod__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)) = N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M2)),N) = zero_zero_nat ) )
      & ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)) != N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M2)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)) ) ) ) ).

% mod_Suc
tff(fact_4826_mod__induct,axiom,
    ! [P2: fun_nat_bool,N: nat,P: nat,M2: nat] :
      ( pp(aa_nat_bool(P2,N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),P))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),P))
         => ( ! [N3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),P))
               => ( pp(aa_nat_bool(P2,N3))
                 => pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N3)),P))) ) )
           => pp(aa_nat_bool(P2,M2)) ) ) ) ) ).

% mod_induct
tff(fact_4827_mod__less__divisor,axiom,
    ! [N: nat,M2: 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(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)),N)) ) ).

% mod_less_divisor
tff(fact_4828_gcd__nat__induct,axiom,
    ! [P2: fun_nat_fun_nat_bool,M2: nat,N: nat] :
      ( ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,M),zero_zero_nat))
     => ( ! [M: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,N3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N3)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,M),N3)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P2,M2),N)) ) ) ).

% gcd_nat_induct
tff(fact_4829_mod__Suc__le__divisor,axiom,
    ! [M2: nat,N: 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,M2),aa_nat_nat(suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_4830_mod__eq__0D,axiom,
    ! [M2: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),D) = zero_zero_nat )
     => ? [Q3: nat] : M2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_4831_mod__geq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),N) ) ) ).

% mod_geq
tff(fact_4832_le__mod__geq,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),N) ) ) ).

% le_mod_geq
tff(fact_4833_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
    <=> ? [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,N),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_4834_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_4835_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(aa_int_fun_int_int(modulo_modulo_int,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4836_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_4837_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(aa_int_fun_int_int(modulo_modulo_int,A),B))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4838_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_4839_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_4840_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_4841_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_4842_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_4843_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_4844_cong__exp__iff__simps_I8_J,axiom,
    ! [M2: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_4845_cong__exp__iff__simps_I8_J,axiom,
    ! [M2: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_4846_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C2: 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)),C2) = 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),C2))),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),C2))),C2)) ).

% div_mult1_eq
tff(fact_4847_div__mult1__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C2))),C2)) ).

% div_mult1_eq
tff(fact_4848_div__mult1__eq,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] : divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),C2) = plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),divide5121882707175180666atural(B,C2)),divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),modulo8411746178871703098atural(B,C2)),C2)) ).

% div_mult1_eq
tff(fact_4849_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C2: 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))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ).

% cancel_div_mod_rules(2)
tff(fact_4850_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) ).

% cancel_div_mod_rules(2)
tff(fact_4851_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_natural,A: code_natural,C2: code_natural] : plus_p4538020629002901425atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),divide5121882707175180666atural(A,B)),modulo8411746178871703098atural(A,B)),C2) = plus_p4538020629002901425atural(A,C2) ).

% cancel_div_mod_rules(2)
tff(fact_4852_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C2: 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))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ).

% cancel_div_mod_rules(1)
tff(fact_4853_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) ).

% cancel_div_mod_rules(1)
tff(fact_4854_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] : plus_p4538020629002901425atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(A,B)),B),modulo8411746178871703098atural(A,B)),C2) = plus_p4538020629002901425atural(A,C2) ).

% cancel_div_mod_rules(1)
tff(fact_4855_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_4856_mod__div__decomp,axiom,
    ! [A: int,B: int] : A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% mod_div_decomp
tff(fact_4857_mod__div__decomp,axiom,
    ! [A: code_natural,B: code_natural] : A = plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(A,B)),B),modulo8411746178871703098atural(A,B)) ).

% mod_div_decomp
tff(fact_4858_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_4859_div__mult__mod__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% div_mult_mod_eq
tff(fact_4860_div__mult__mod__eq,axiom,
    ! [A: code_natural,B: code_natural] : plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(A,B)),B),modulo8411746178871703098atural(A,B)) = A ).

% div_mult_mod_eq
tff(fact_4861_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_4862_mod__div__mult__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_4863_mod__div__mult__eq,axiom,
    ! [A: code_natural,B: code_natural] : plus_p4538020629002901425atural(modulo8411746178871703098atural(A,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(A,B)),B)) = A ).

% mod_div_mult_eq
tff(fact_4864_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_4865_mod__mult__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = A ).

% mod_mult_div_eq
tff(fact_4866_mod__mult__div__eq,axiom,
    ! [A: code_natural,B: code_natural] : plus_p4538020629002901425atural(modulo8411746178871703098atural(A,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),divide5121882707175180666atural(A,B))) = A ).

% mod_mult_div_eq
tff(fact_4867_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_4868_mult__div__mod__eq,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% mult_div_mod_eq
tff(fact_4869_mult__div__mod__eq,axiom,
    ! [B: code_natural,A: code_natural] : plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),divide5121882707175180666atural(A,B)),modulo8411746178871703098atural(A,B)) = A ).

% mult_div_mod_eq
tff(fact_4870_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_4871_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_div_mult_eq_mod
tff(fact_4872_minus__div__mult__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] : minus_7197305767214868737atural(A,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(A,B)),B)) = modulo8411746178871703098atural(A,B) ).

% minus_div_mult_eq_mod
tff(fact_4873_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_4874_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4875_minus__mod__eq__div__mult,axiom,
    ! [A: code_natural,B: code_natural] : minus_7197305767214868737atural(A,modulo8411746178871703098atural(A,B)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(A,B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4876_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_4877_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4878_minus__mod__eq__mult__div,axiom,
    ! [A: code_natural,B: code_natural] : minus_7197305767214868737atural(A,modulo8411746178871703098atural(A,B)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),divide5121882707175180666atural(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_4879_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_4880_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_mult_div_eq_mod
tff(fact_4881_minus__mult__div__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] : minus_7197305767214868737atural(A,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),divide5121882707175180666atural(A,B))) = modulo8411746178871703098atural(A,B) ).

% minus_mult_div_eq_mod
tff(fact_4882_mod__le__divisor,axiom,
    ! [N: nat,M2: 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(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)),N)) ) ).

% mod_le_divisor
tff(fact_4883_div__less__mono,axiom,
    ! [A4: nat,B5: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A4),B5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),N) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B5),N) = 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,A4),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B5),N))) ) ) ) ) ).

% div_less_mono
tff(fact_4884_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
       => ? [Q3: 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,N),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_4885_mod__eq__nat2E,axiom,
    ! [M2: nat,Q2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ~ ! [S: nat] : N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S)) ) ) ).

% mod_eq_nat2E
tff(fact_4886_mod__eq__nat1E,axiom,
    ! [M2: nat,Q2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ~ ! [S: nat] : M2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S)) ) ) ).

% mod_eq_nat1E
tff(fact_4887_mod__mult2__eq,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)) ).

% mod_mult2_eq
tff(fact_4888_div__mod__decomp,axiom,
    ! [A4: nat,N: nat] : A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),N)),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),N)) ).

% div_mod_decomp
tff(fact_4889_modulo__nat__def,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),N)) ).

% modulo_nat_def
tff(fact_4890_mod__mult2__eq_H,axiom,
    ! [A: code_natural,M2: nat,N: nat] : modulo8411746178871703098atural(A,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,semiri3763490453095760265atural(M2)),semiri3763490453095760265atural(N))) = plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,semiri3763490453095760265atural(M2)),modulo8411746178871703098atural(divide5121882707175180666atural(A,semiri3763490453095760265atural(M2)),semiri3763490453095760265atural(N))),modulo8411746178871703098atural(A,semiri3763490453095760265atural(M2))) ).

% mod_mult2_eq'
tff(fact_4891_mod__mult2__eq_H,axiom,
    ! [A: int,M2: nat,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N))) = 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,M2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_nat_int(semiri1314217659103216013at_int,N)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(semiri1314217659103216013at_int,M2))) ).

% mod_mult2_eq'
tff(fact_4892_mod__mult2__eq_H,axiom,
    ! [A: nat,M2: nat,N: 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,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = 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,M2)),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,M2))),aa_nat_nat(semiri1316708129612266289at_nat,N)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M2))) ).

% mod_mult2_eq'
tff(fact_4893_split__mod,axiom,
    ! [P2: fun_nat_bool,M2: nat,N: nat] :
      ( pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P2,M2)) )
        & ( ( N != zero_zero_nat )
         => ! [I: nat,J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),N))
             => ( ( M2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I)),J) )
               => pp(aa_nat_bool(P2,J)) ) ) ) ) ) ).

% split_mod
tff(fact_4894_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) = 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)),C2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4895_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4896_Suc__times__mod__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2))
     => ( 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,M2),N))),M2) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_4897_nth__rotate1,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_VEBT_VEBT(rotate1_VEBT_VEBT(Xs),N) = nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),size_s6755466524823107622T_VEBT(Xs))) ) ) ).

% nth_rotate1
tff(fact_4898_nth__rotate1,axiom,
    ! [N: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(nth_bool(rotate1_bool(Xs)),N))
      <=> pp(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),size_size_list_bool(Xs)))) ) ) ).

% nth_rotate1
tff(fact_4899_nth__rotate1,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( aa_nat_nat(nth_nat(rotate1_nat(Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),aa_list_nat_nat(size_size_list_nat,Xs))) ) ) ).

% nth_rotate1
tff(fact_4900_nth__rotate1,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => ( aa_nat_int(nth_int(rotate1_int(Xs)),N) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),size_size_list_int(Xs))) ) ) ).

% nth_rotate1
tff(fact_4901_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_4902_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4903_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_4904_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_4905_bits__stable__imp__add__self,axiom,
    ! [A: code_natural] :
      ( ( divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = A )
     => ( plus_p4538020629002901425atural(A,modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))) = zero_z2226904508553997617atural ) ) ).

% bits_stable_imp_add_self
tff(fact_4906_verit__le__mono__div,axiom,
    ! [A4: nat,B5: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A4),B5))
     => ( 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(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),N)),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,B5),N)),zero_zero_nat)),one_one_nat),zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B5),N))) ) ) ).

% verit_le_mono_div
tff(fact_4907_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_4908_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B))
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4909_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( 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))),M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = 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,N),M2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4910_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4911_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: code_natural] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),M2)),power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,modulo8411746178871703098atural(A,power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))),power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),M2)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4912_mod__double__modulus,axiom,
    ! [M2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( 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))),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M2) )
          | ( 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))),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M2)),M2) ) ) ) ) ).

% mod_double_modulus
tff(fact_4913_mod__double__modulus,axiom,
    ! [M2: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M2))
     => ( 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(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))),M2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),M2) )
          | ( aa_int_int(aa_int_fun_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))),M2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),M2)),M2) ) ) ) ) ).

% mod_double_modulus
tff(fact_4914_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_4915_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4916_unset__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,N)),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,N),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_4917_unset__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : bit_se7083795435491715335atural(aa_nat_nat(suc,N),A) = plus_p4538020629002901425atural(modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(aa_num_num(bit0,one))),bit_se7083795435491715335atural(N,divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4918_unset__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4919_set__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,N)),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,N),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_4920_set__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : bit_se1617098188084679374atural(aa_nat_nat(suc,N),A) = plus_p4538020629002901425atural(modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(aa_num_num(bit0,one))),bit_se1617098188084679374atural(N,divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4921_set__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se7879613467334960850it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4922_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( finite_card_complex(collect_complex(aTP_Lamm_bl(N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_4923_card__nth__roots,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_bm(C2),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_4924_product__nth,axiom,
    ! [N: nat,Xs: list_Code_integer,Ys: list_Code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3445333598471063425nteger(Xs)),size_s3445333598471063425nteger(Ys))))
     => ( nth_Pr2304437835452373666nteger(produc8792966785426426881nteger(Xs,Ys),N) = produc1086072967326762835nteger(aa_nat_Code_integer(nth_Code_integer(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s3445333598471063425nteger(Ys))),aa_nat_Code_integer(nth_Code_integer(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s3445333598471063425nteger(Ys)))) ) ) ).

% product_nth
tff(fact_4925_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys))))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys),N) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_4926_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_bool(Ys))))
     => ( nth_Pr1289831921885633937T_bool(produc5466675335756156456T_bool(Xs,Ys),N) = produc3035007008238225530T_bool(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_bool(Ys))),aa_nat_bool(nth_bool(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_bool(Ys)))) ) ) ).

% product_nth
tff(fact_4927_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),aa_list_nat_nat(size_size_list_nat,Ys))))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys),N) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_list_nat_nat(size_size_list_nat,Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_list_nat_nat(size_size_list_nat,Ys)))) ) ) ).

% product_nth
tff(fact_4928_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys))))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys),N) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_4929_product__nth,axiom,
    ! [N: nat,Xs: list_bool,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_s6755466524823107622T_VEBT(Ys))))
     => ( nth_Pr3250499561091541003T_VEBT(produc2282853104913622818T_VEBT(Xs,Ys),N) = produc9074556814250467700T_VEBT(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_4930_product__nth,axiom,
    ! [N: nat,Xs: list_bool,Ys: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_bool(Ys))))
     => ( nth_Pr2856333732135533678l_bool(product_bool_bool(Xs,Ys),N) = produc8297744024737662807l_bool(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_bool(Ys))),aa_nat_bool(nth_bool(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_bool(Ys)))) ) ) ).

% product_nth
tff(fact_4931_product__nth,axiom,
    ! [N: nat,Xs: list_bool,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),aa_list_nat_nat(size_size_list_nat,Ys))))
     => ( nth_Pr2186422209488468047ol_nat(product_bool_nat(Xs,Ys),N) = produc8874003862241262736ol_nat(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_list_nat_nat(size_size_list_nat,Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_list_nat_nat(size_size_list_nat,Ys)))) ) ) ).

% product_nth
tff(fact_4932_product__nth,axiom,
    ! [N: nat,Xs: list_bool,Ys: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_int(Ys))))
     => ( nth_Pr7231943226834047147ol_int(product_bool_int(Xs,Ys),N) = produc8871513391732212460ol_int(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_4933_product__nth,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),size_s6755466524823107622T_VEBT(Ys))))
     => ( nth_Pr744662078594809490T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys),N) = produc599794634098209291T_VEBT(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_4934_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_4935_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_4936_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_4937_VEBT__internal_Oinsert_H_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) )
             => ( ( Y = vEBT_vebt_insert(vEBT_Leaf(A3,B3),Xa2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2)) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
               => ( ( ( 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))),Deg2)),Xa2))
                     => ( Y = vEBT_Node(Info,Deg2,TreeList,Summary2) ) )
                    & ( ~ 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))),Deg2)),Xa2))
                     => ( Y = vEBT_vebt_insert(vEBT_Node(Info,Deg2,TreeList,Summary2),Xa2) ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info,Deg2,TreeList,Summary2),Xa2)) ) ) ) ) ) ).

% VEBT_internal.insert'.pelims
tff(fact_4938_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,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw),Xa2)) ) )
           => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz2) )
                 => ( ~ pp(Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz2),Xa2)) ) )
             => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V),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(V),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_4939_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,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw),Xa2)) )
           => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz2),Xa2)) )
             => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V),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_4940_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,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw),Xa2)) ) )
           => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,S) )
                 => ( ( 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,V)),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,V)),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,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,Xa2),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(TreeList))) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,S),Xa2)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_4941_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 ) ) ) ) ) )
         => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,S) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,S),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,V)),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,V)),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,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,Xa2),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(TreeList))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_4942_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,Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw),Xa2)) )
           => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,S) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,S),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,V)),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,V)),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,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,Xa2),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(TreeList))) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_4943_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_4944_zmod__numeral__Bit0,axiom,
    ! [V2: num,W2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V2))),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(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V2)),aa_num_int(numeral_numeral_int,W2))) ).

% zmod_numeral_Bit0
tff(fact_4945_zmod__le__nonneg__dividend,axiom,
    ! [M2: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M2),K2)),M2)) ) ).

% zmod_le_nonneg_dividend
tff(fact_4946_zmod__eq__0__iff,axiom,
    ! [M2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M2),D) = zero_zero_int )
    <=> ? [Q5: int] : M2 = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q5) ) ).

% zmod_eq_0_iff
tff(fact_4947_zmod__eq__0D,axiom,
    ! [M2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M2),D) = zero_zero_int )
     => ? [Q3: int] : M2 = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q3) ) ).

% zmod_eq_0D
tff(fact_4948_zmod__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zmod_int
tff(fact_4949_mod__int__unique,axiom,
    ! [K2: int,L: int,Q2: int,R: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K2,L),product_Pair_int_int(Q2,R)))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = R ) ) ).

% mod_int_unique
tff(fact_4950_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(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,A),B))) ) ) ).

% neg_mod_conj
tff(fact_4951_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(aa_int_fun_int_int(modulo_modulo_int,A),B)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B)) ) ) ).

% pos_mod_conj
tff(fact_4952_zmod__trivial__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I2),K2) = I2 )
    <=> ( ( K2 = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),K2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),I2)) ) ) ) ).

% zmod_trivial_iff
tff(fact_4953_div__mod__decomp__int,axiom,
    ! [A4: int,N: int] : A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),N)),N)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),N)) ).

% div_mod_decomp_int
tff(fact_4954_eucl__rel__int,axiom,
    ! [K2: int,L: int] : pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K2,L),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L)))) ).

% eucl_rel_int
tff(fact_4955_mod__pos__geq,axiom,
    ! [L: int,K2: 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),K2))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_4956_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),B))
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R ) ) ) ) ).

% int_mod_pos_eq
tff(fact_4957_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R))
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R ) ) ) ) ).

% int_mod_neg_eq
tff(fact_4958_split__zmod,axiom,
    ! [P2: fun_int_bool,N: int,K2: int] :
      ( pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K2)))
    <=> ( ( ( K2 = zero_zero_int )
         => pp(aa_int_bool(P2,N)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
         => ! [I: int,J: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),K2))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J) ) )
             => pp(aa_int_bool(P2,J)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
         => ! [I: int,J: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),J))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),zero_zero_int))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J) ) )
             => pp(aa_int_bool(P2,J)) ) ) ) ) ).

% split_zmod
tff(fact_4959_zmod__zmult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C2))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_4960_split__pos__lemma,axiom,
    ! [K2: int,P2: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P2,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K2)))
      <=> ! [I: int,J: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),K2))
              & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P2,I),J)) ) ) ) ).

% split_pos_lemma
tff(fact_4961_split__neg__lemma,axiom,
    ! [K2: int,P2: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P2,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K2)))
      <=> ! [I: int,J: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),J))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),zero_zero_int))
              & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P2,I),J)) ) ) ) ).

% split_neg_lemma
tff(fact_4962_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(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_4963_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(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_4964_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)) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy),Xa2)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,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,Va2,Vb2),Xa2)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V: 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,V),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,V)),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,V)),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,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,Xa2),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(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,V),TreeList,Vc2),Xa2)) ) )
               => ~ ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd) )
                     => ( ( 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,V)),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,V)),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,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,Xa2),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(TreeList))) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd),Xa2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_4965_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)) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy),Xa2)) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2),Xa2))
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V: 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,V),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,V),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,V)),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,V)),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,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,Xa2),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(TreeList))) ) ) ) )
               => ~ ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd),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,V)),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,V)),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,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,Xa2),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(TreeList))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_4966_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,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2),Xa2))
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V: 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,V),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,V),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,V)),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,V)),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,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,Xa2),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(TreeList))) ) ) ) )
           => ~ ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd),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,V)),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,V)),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,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,Xa2),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(TreeList))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_4967_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_4968_lemma__termdiff3,axiom,
    ! [H: real,Z2: real,K5: real,N: 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(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(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Z2),H)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),N))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),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,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_4969_lemma__termdiff3,axiom,
    ! [H: complex,Z2: complex,K5: real,N: 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)),N)),aa_nat_complex(power_power_complex(Z2),N))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(power_power_complex(Z2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),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,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_4970_flip__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,N)),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,N),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_4971_flip__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4972_flip__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : bit_se168947363167071951atural(aa_nat_nat(suc,N),A) = plus_p4538020629002901425atural(modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(aa_num_num(bit0,one))),bit_se168947363167071951atural(N,divide5121882707175180666atural(A,numera5444537566228673987atural(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4973_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_4974_norm__mult__numeral1,axiom,
    ! [W2: num,A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(real_V7735802525324610683m_real,A)) ).

% norm_mult_numeral1
tff(fact_4975_norm__mult__numeral1,axiom,
    ! [W2: num,A: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% norm_mult_numeral1
tff(fact_4976_norm__mult__numeral2,axiom,
    ! [A: real,W2: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W2)) ).

% norm_mult_numeral2
tff(fact_4977_norm__mult__numeral2,axiom,
    ! [A: complex,W2: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_num_real(numeral_numeral_real,W2)) ).

% norm_mult_numeral2
tff(fact_4978_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_4979_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_4980_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_4981_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_4982_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_4983_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_4984_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_4985_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,X) = zero_zero_real )
    <=> ( X = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_4986_norm__power__diff,axiom,
    ! [Z2: real,W2: real,M2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z2)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W2)),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),M2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Z2),W2))))) ) ) ).

% norm_power_diff
tff(fact_4987_norm__power__diff,axiom,
    ! [Z2: complex,W2: complex,M2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W2)),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Z2),M2)),aa_nat_complex(power_power_complex(W2),M2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Z2),W2))))) ) ) ).

% norm_power_diff
tff(fact_4988_norm__mult,axiom,
    ! [X: real,Y: 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,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)) ).

% norm_mult
tff(fact_4989_norm__mult,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)) ).

% norm_mult
tff(fact_4990_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ) ) ).

% nonzero_norm_divide
tff(fact_4991_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ) ) ).

% nonzero_norm_divide
tff(fact_4992_power__eq__imp__eq__norm,axiom,
    ! [W2: real,N: nat,Z2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_real_real(real_V7735802525324610683m_real,W2) = aa_real_real(real_V7735802525324610683m_real,Z2) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_4993_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,N: nat,Z2: complex] :
      ( ( aa_nat_complex(power_power_complex(W2),N) = aa_nat_complex(power_power_complex(Z2),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_complex_real(real_V1022390504157884413omplex,W2) = aa_complex_real(real_V1022390504157884413omplex,Z2) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_4994_norm__mult__less,axiom,
    ! [X: real,R: real,Y: real,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S2))
       => 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,R),S2))) ) ) ).

% norm_mult_less
tff(fact_4995_norm__mult__less,axiom,
    ! [X: complex,R: real,Y: complex,S2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S2))
       => 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,R),S2))) ) ) ).

% norm_mult_less
tff(fact_4996_norm__mult__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(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)))) ).

% norm_mult_ineq
tff(fact_4997_norm__mult__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(times_times_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))) ).

% norm_mult_ineq
tff(fact_4998_power__eq__1__iff,axiom,
    ! [W2: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),N) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W2) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_4999_power__eq__1__iff,axiom,
    ! [W2: complex,N: nat] :
      ( ( aa_nat_complex(power_power_complex(W2),N) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W2) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_5000_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_5001_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_5002_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_re6978948838062957830nteger,Y: fun_re6978948838062957830nteger] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun2607813867624603264l_bool(aTP_Lamm_bn(I5),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun2607813867624603264l_bool(aTP_Lamm_bn(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun2607813867624603264l_bool(aa_fun1301660912713367635l_bool(aTP_Lamm_bo(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5003_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_Code_integer,Y: fun_nat_Code_integer] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1752071697927428280t_bool(aTP_Lamm_bp(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1752071697927428280t_bool(aTP_Lamm_bp(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1752071697927428280t_bool(aa_fun4813323487062687079t_bool(aTP_Lamm_bq(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5004_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_Code_integer,Y: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun3610984093196804608t_bool(aTP_Lamm_br(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun3610984093196804608t_bool(aTP_Lamm_br(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun3610984093196804608t_bool(aa_fun2519502641837456723t_bool(aTP_Lamm_bs(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5005_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_co8153928105989009288nteger,Y: fun_co8153928105989009288nteger] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun6924306962011675644x_bool(aTP_Lamm_bt(I5),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun6924306962011675644x_bool(aTP_Lamm_bt(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun6924306962011675644x_bool(aa_fun4664215553580952141x_bool(aTP_Lamm_bu(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5006_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_complex,Y: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_bv(I5),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_bv(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_bw(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5007_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_bx(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_bx(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_by(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5008_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_bz(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_bz(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_ca(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5009_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_cb(I5),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_cb(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_cc(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5010_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_real,Y: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_cd(I5),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_cd(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_ce(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5011_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_cf(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_cf(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_cg(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5012_sum__gp,axiom,
    ! [N: nat,M2: nat,X: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
       => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
       => ( ( ( X = one_one_complex )
           => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = 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,N),one_one_nat)),M2)) ) )
          & ( ( X != one_one_complex )
           => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M2)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N)))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_5013_sum__gp,axiom,
    ! [N: nat,M2: nat,X: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
       => ( ( ( X = one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = 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,N),one_one_nat)),M2)) ) )
          & ( ( X != one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),M2)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_5014_sum__gp,axiom,
    ! [N: nat,M2: nat,X: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
       => ( ( ( X = one_one_real )
           => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = 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,N),one_one_nat)),M2)) ) )
          & ( ( X != one_one_real )
           => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N)))),aa_real_real(minus_minus_real(one_one_real),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_5015_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_complex,Y: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_ch(I5),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_ch(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_ci(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5016_sum_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_cj(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_cj(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_ck(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5017_sum_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_cl(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_cl(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_cm(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5018_sum_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_cn(I5),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_cn(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_co(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5019_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_real,Y: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_cp(I5),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_cp(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_cq(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5020_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_cr(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_cr(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_cs(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5021_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_ct(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ct(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_cu(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5022_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_cv(I5),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_cv(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_cw(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5023_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_rat,Y: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_cx(I5),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_cx(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_cy(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5024_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_cz(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_cz(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_da(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5025_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(aa_real_fun_nat_real(aTP_Lamm_db,Z2)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(one_one_real),Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% geometric_deriv_sums
tff(fact_5026_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_dc(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_5027_pochhammer__double,axiom,
    ! [Z2: complex,N: 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))),N)) = 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))),N)))),aa_nat_complex(comm_s2602460028002588243omplex(Z2),N))),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))))),N)) ).

% pochhammer_double
tff(fact_5028_pochhammer__double,axiom,
    ! [Z2: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_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))),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z2),N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_5029_pochhammer__double,axiom,
    ! [Z2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = 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))),N)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z2),N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(Z2),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),N)) ).

% pochhammer_double
tff(fact_5030_dvd__0__right,axiom,
    ! [A: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),zero_zero_complex)) ).

% dvd_0_right
tff(fact_5031_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_5032_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_5033_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_5034_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_5035_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,zero_zero_complex),A))
    <=> ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
tff(fact_5036_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_5037_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_5038_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_5039_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_5040_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_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_5041_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(aa_rat_fun_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_5042_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_5043_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_5044_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_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_5045_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(aa_rat_fun_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_5046_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_5047_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_5048_div__dvd__div,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => ( 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,C2),A)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C2)) ) ) ) ).

% div_dvd_div
tff(fact_5049_div__dvd__div,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),A)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C2)) ) ) ) ).

% div_dvd_div
tff(fact_5050_nat__dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),one_one_nat))
    <=> ( M2 = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_5051_sum_Oneutral__const,axiom,
    ! [A4: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_al),A4) = zero_zero_nat ).

% sum.neutral_const
tff(fact_5052_sum_Oneutral__const,axiom,
    ! [A4: set_int] : groups4538972089207619220nt_int(aTP_Lamm_am,A4) = zero_zero_int ).

% sum.neutral_const
tff(fact_5053_sum_Oneutral__const,axiom,
    ! [A4: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_dd),A4) = zero_zero_real ).

% sum.neutral_const
tff(fact_5054_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C2: 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,C2),A)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C2)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_5055_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C2: 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,C2),A)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C2)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_5056_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C2: 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),C2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C2)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_5057_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C2: 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),C2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C2)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_5058_dvd__mult__cancel__right,axiom,
    ! [A: complex,C2: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),C2)),aa_complex_complex(times_times_complex(B),C2)))
    <=> ( ( C2 = zero_zero_complex )
        | pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_5059_dvd__mult__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)))
    <=> ( ( C2 = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_5060_dvd__mult__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)))
    <=> ( ( C2 = zero_zero_rat )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_5061_dvd__mult__cancel__right,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)))
    <=> ( ( C2 = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_5062_dvd__mult__cancel__left,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(C2),A)),aa_complex_complex(times_times_complex(C2),B)))
    <=> ( ( C2 = zero_zero_complex )
        | pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_5063_dvd__mult__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)))
    <=> ( ( C2 = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_5064_dvd__mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)))
    <=> ( ( C2 = zero_zero_rat )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_5065_dvd__mult__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)))
    <=> ( ( C2 = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_5066_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5067_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5068_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C2: 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,C2),A))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5069_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C2: 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,C2),A))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5070_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5071_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5072_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C2: 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,C2),A)),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5073_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C2: 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,C2),A)),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5074_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer)) ) ) ).

% unit_prod
tff(fact_5075_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_5076_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_5077_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_5078_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(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_5079_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_5080_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(aa_int_fun_int_int(divide_divide_int,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_5081_div__add,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),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)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2)) ) ) ) ).

% div_add
tff(fact_5082_div__add,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2)) ) ) ) ).

% div_add
tff(fact_5083_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),one_one_Code_integer)) ) ) ).

% unit_div
tff(fact_5084_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_5085_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(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int)) ) ) ).

% unit_div
tff(fact_5086_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)),one_one_Code_integer)) ) ).

% unit_div_1_unit
tff(fact_5087_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_5088_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(aa_int_fun_int_int(divide_divide_int,one_one_int),A)),one_one_int)) ) ).

% unit_div_1_unit
tff(fact_5089_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( divide6298287555418463151nteger(one_one_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_5090_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_5091_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(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_5092_sum_Oempty,axiom,
    ! [G2: fun_nat_complex] : aa_set_nat_complex(groups2073611262835488442omplex(G2),bot_bot_set_nat) = zero_zero_complex ).

% sum.empty
tff(fact_5093_sum_Oempty,axiom,
    ! [G2: fun_nat_rat] : groups2906978787729119204at_rat(G2,bot_bot_set_nat) = zero_zero_rat ).

% sum.empty
tff(fact_5094_sum_Oempty,axiom,
    ! [G2: fun_nat_int] : groups3539618377306564664at_int(G2,bot_bot_set_nat) = zero_zero_int ).

% sum.empty
tff(fact_5095_sum_Oempty,axiom,
    ! [G2: fun_int_complex] : groups3049146728041665814omplex(G2,bot_bot_set_int) = zero_zero_complex ).

% sum.empty
tff(fact_5096_sum_Oempty,axiom,
    ! [G2: fun_int_real] : groups8778361861064173332t_real(G2,bot_bot_set_int) = zero_zero_real ).

% sum.empty
tff(fact_5097_sum_Oempty,axiom,
    ! [G2: fun_int_rat] : groups3906332499630173760nt_rat(G2,bot_bot_set_int) = zero_zero_rat ).

% sum.empty
tff(fact_5098_sum_Oempty,axiom,
    ! [G2: fun_int_nat] : groups4541462559716669496nt_nat(G2,bot_bot_set_int) = zero_zero_nat ).

% sum.empty
tff(fact_5099_sum_Oempty,axiom,
    ! [G2: fun_real_complex] : groups5754745047067104278omplex(G2,bot_bot_set_real) = zero_zero_complex ).

% sum.empty
tff(fact_5100_sum_Oempty,axiom,
    ! [G2: fun_real_real] : groups8097168146408367636l_real(G2,bot_bot_set_real) = zero_zero_real ).

% sum.empty
tff(fact_5101_sum_Oempty,axiom,
    ! [G2: fun_real_rat] : groups1300246762558778688al_rat(G2,bot_bot_set_real) = zero_zero_rat ).

% sum.empty
tff(fact_5102_sum_Oinfinite,axiom,
    ! [A4: set_nat,G2: fun_nat_complex] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( aa_set_nat_complex(groups2073611262835488442omplex(G2),A4) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_5103_sum_Oinfinite,axiom,
    ! [A4: set_int,G2: fun_int_complex] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups3049146728041665814omplex(G2,A4) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_5104_sum_Oinfinite,axiom,
    ! [A4: set_complex,G2: fun_complex_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups7754918857620584856omplex(G2,A4) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_5105_sum_Oinfinite,axiom,
    ! [A4: set_int,G2: fun_int_real] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups8778361861064173332t_real(G2,A4) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_5106_sum_Oinfinite,axiom,
    ! [A4: set_complex,G2: fun_complex_real] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5808333547571424918x_real(G2,A4) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_5107_sum_Oinfinite,axiom,
    ! [A4: set_nat,G2: fun_nat_rat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( groups2906978787729119204at_rat(G2,A4) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_5108_sum_Oinfinite,axiom,
    ! [A4: set_int,G2: fun_int_rat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups3906332499630173760nt_rat(G2,A4) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_5109_sum_Oinfinite,axiom,
    ! [A4: set_complex,G2: fun_complex_rat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5058264527183730370ex_rat(G2,A4) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_5110_sum_Oinfinite,axiom,
    ! [A4: set_int,G2: fun_int_nat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups4541462559716669496nt_nat(G2,A4) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_5111_sum_Oinfinite,axiom,
    ! [A4: set_complex,G2: fun_complex_nat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5693394587270226106ex_nat(G2,A4) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_5112_sum__eq__0__iff,axiom,
    ! [F3: set_int,F2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( ( groups4541462559716669496nt_nat(F2,F3) = zero_zero_nat )
      <=> ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),F3))
           => ( aa_int_nat(F2,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5113_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( ( groups5693394587270226106ex_nat(F2,F3) = zero_zero_nat )
      <=> ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),F3))
           => ( aa_complex_nat(F2,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5114_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F2: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F2),F3) = zero_zero_nat )
      <=> ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),F3))
           => ( aa_nat_nat(F2,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5115_div__diff,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2)) ) ) ) ).

% div_diff
tff(fact_5116_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_5117_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(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_5118_dvd__imp__mod__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( dvd_dvd_Code_natural(A,B)
     => ( modulo8411746178871703098atural(B,A) = zero_z2226904508553997617atural ) ) ).

% dvd_imp_mod_0
tff(fact_5119_dvd__1__left,axiom,
    ! [K2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K2)) ).

% dvd_1_left
tff(fact_5120_dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( M2 = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_5121_nat__mult__dvd__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)))
    <=> ( ( K2 = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_5122_pochhammer__0,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),zero_zero_nat) = one_one_Code_integer ).

% pochhammer_0
tff(fact_5123_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_5124_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_5125_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_5126_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_5127_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( pp(aa_set_real_bool(member_real(A),S3))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_de(A),B),S3) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S3))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_de(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_5128_sum_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))
         => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_df(A),B)),S3) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_df(A),B)),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_5129_sum_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))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_dg(A),B),S3) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_dg(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_5130_sum_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))
         => ( groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_dh(A),B),S3) = aa_complex_complex(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_dh(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_5131_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( pp(aa_set_real_bool(member_real(A),S3))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_di(A),B),S3) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S3))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_di(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5132_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_dj(A),B),S3) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_dj(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5133_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_dk(A),B),S3) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_dk(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_5134_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( pp(aa_set_real_bool(member_real(A),S3))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_dl(A),B),S3) = aa_real_rat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S3))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_dl(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_5135_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_dm(A),B),S3) = aa_nat_rat(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_dm(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_5136_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_dn(A),B),S3) = aa_int_rat(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_dn(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_5137_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( pp(aa_set_real_bool(member_real(A),S3))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_do(A),B),S3) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S3))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_do(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_5138_sum_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))
         => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_dp(A),B)),S3) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_dp(A),B)),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_5139_sum_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))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_dq(A),B),S3) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_dq(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_5140_sum_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))
         => ( groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_dr(A),B),S3) = aa_complex_complex(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_dr(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_5141_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( pp(aa_set_real_bool(member_real(A),S3))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ds(A),B),S3) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S3))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ds(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5142_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_dt(A),B),S3) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_dt(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5143_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_du(A),B),S3) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_du(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_5144_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( pp(aa_set_real_bool(member_real(A),S3))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_dv(A),B),S3) = aa_real_rat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S3))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_dv(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_5145_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_dw(A),B),S3) = aa_nat_rat(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_dw(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_5146_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_dx(A),B),S3) = aa_int_rat(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_dx(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_5147_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(one_one_Code_integer,A)) = divide6298287555418463151nteger(B,A) ) ) ).

% unit_mult_div_div
tff(fact_5148_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_5149_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(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% unit_mult_div_div
tff(fact_5150_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_5151_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_5152_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(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_5153_pow__divides__pow__iff,axiom,
    ! [N: nat,A: nat,B: 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(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ) ).

% pow_divides_pow_iff
tff(fact_5154_pow__divides__pow__iff,axiom,
    ! [N: nat,A: int,B: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% pow_divides_pow_iff
tff(fact_5155_sum__constant,axiom,
    ! [Y: rat,A4: set_nat] : groups2906978787729119204at_rat(aTP_Lamm_dy(Y),A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A4))),Y) ).

% sum_constant
tff(fact_5156_sum__constant,axiom,
    ! [Y: rat,A4: set_complex] : groups5058264527183730370ex_rat(aTP_Lamm_dz(Y),A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_complex(A4))),Y) ).

% sum_constant
tff(fact_5157_sum__constant,axiom,
    ! [Y: rat,A4: set_int] : groups3906332499630173760nt_rat(aTP_Lamm_ea(Y),A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_int(A4))),Y) ).

% sum_constant
tff(fact_5158_sum__constant,axiom,
    ! [Y: real,A4: set_complex] : groups5808333547571424918x_real(aTP_Lamm_eb(Y),A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_complex(A4))),Y) ).

% sum_constant
tff(fact_5159_sum__constant,axiom,
    ! [Y: real,A4: set_int] : groups8778361861064173332t_real(aTP_Lamm_ec(Y),A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_int(A4))),Y) ).

% sum_constant
tff(fact_5160_sum__constant,axiom,
    ! [Y: int,A4: set_nat] : groups3539618377306564664at_int(aTP_Lamm_ed(Y),A4) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(finite_card_nat,A4))),Y) ).

% sum_constant
tff(fact_5161_sum__constant,axiom,
    ! [Y: int,A4: set_complex] : groups5690904116761175830ex_int(aTP_Lamm_ee(Y),A4) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_complex(A4))),Y) ).

% sum_constant
tff(fact_5162_sum__constant,axiom,
    ! [Y: nat,A4: set_complex] : groups5693394587270226106ex_nat(aTP_Lamm_ef(Y),A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_complex(A4))),Y) ).

% sum_constant
tff(fact_5163_sum__constant,axiom,
    ! [Y: nat,A4: set_int] : groups4541462559716669496nt_nat(aTP_Lamm_eg(Y),A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_int(A4))),Y) ).

% sum_constant
tff(fact_5164_sum__constant,axiom,
    ! [Y: nat,A4: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_eh,Y)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(finite_card_nat,A4))),Y) ).

% sum_constant
tff(fact_5165_even__mult__iff,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(times_times_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_mult_iff
tff(fact_5166_even__mult__iff,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(times_times_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_mult_iff
tff(fact_5167_even__Suc__Suc__iff,axiom,
    ! [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))),aa_nat_nat(suc,aa_nat_nat(suc,N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_Suc_Suc_iff
tff(fact_5168_even__Suc,axiom,
    ! [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))),aa_nat_nat(suc,N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_Suc
tff(fact_5169_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,X: complex] :
      ( pp(aa_complex_bool(sums_complex(aTP_Lamm_ei(A)),X))
    <=> ( aa_nat_complex(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_5170_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,X: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_ej(A)),X))
    <=> ( aa_nat_real(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_5171_even__Suc__div__two,axiom,
    ! [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))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_5172_odd__Suc__div__two,axiom,
    ! [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))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_5173_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_complex(groups2073611262835488442omplex(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_complex(groups2073611262835488442omplex(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_complex_complex(plus_plus_complex(aa_set_nat_complex(groups2073611262835488442omplex(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_complex(G2,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_5174_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_rat(G2,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_5175_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups3539618377306564664at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = zero_zero_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups3539618377306564664at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_int(G2,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_5176_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_nat(G2,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_5177_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_real(G2,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_5178_sum__zero__power,axiom,
    ! [A4: set_nat,C2: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A4))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_ei(C2)),A4) = aa_nat_complex(C2,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A4))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_ei(C2)),A4) = zero_zero_complex ) ) ) ).

% sum_zero_power
tff(fact_5179_sum__zero__power,axiom,
    ! [A4: set_nat,C2: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A4))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_ek(C2),A4) = aa_nat_rat(C2,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A4))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_ek(C2),A4) = zero_zero_rat ) ) ) ).

% sum_zero_power
tff(fact_5180_sum__zero__power,axiom,
    ! [A4: set_nat,C2: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A4))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ej(C2)),A4) = aa_nat_real(C2,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A4))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ej(C2)),A4) = zero_zero_real ) ) ) ).

% sum_zero_power
tff(fact_5181_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(aa_real_fun_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_5182_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_5183_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_5184_even__power,axiom,
    ! [A: nat,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))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
    <=> ( 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),N)) ) ) ).

% even_power
tff(fact_5185_even__power,axiom,
    ! [A: int,N: 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),N)))
    <=> ( 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),N)) ) ) ).

% even_power
tff(fact_5186_power__less__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),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))),N))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% power_less_zero_eq
tff(fact_5187_power__less__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),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))),N))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).

% power_less_zero_eq
tff(fact_5188_power__less__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),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))),N))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% power_less_zero_eq
tff(fact_5189_power__less__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_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_5190_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_5191_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_5192_odd__Suc__minus__one,axiom,
    ! [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))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_5193_sum__zero__power_H,axiom,
    ! [A4: set_nat,C2: fun_nat_complex,D: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A4))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_el(C2),D)),A4) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(C2,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A4))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_el(C2),D)),A4) = zero_zero_complex ) ) ) ).

% sum_zero_power'
tff(fact_5194_sum__zero__power_H,axiom,
    ! [A4: set_nat,C2: fun_nat_rat,D: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A4))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_em(C2),D),A4) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(C2,zero_zero_nat)),aa_nat_rat(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A4))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_em(C2),D),A4) = zero_zero_rat ) ) ) ).

% sum_zero_power'
tff(fact_5195_sum__zero__power_H,axiom,
    ! [A4: set_nat,C2: fun_nat_real,D: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A4))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_en(C2),D)),A4) = aa_real_real(divide_divide_real(aa_nat_real(C2,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A4))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_en(C2),D)),A4) = zero_zero_real ) ) ) ).

% sum_zero_power'
tff(fact_5196_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),one_one_Code_integer) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_5197_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_5198_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(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_5199_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N),one_one_Code_integer)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5200_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [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))),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))),N)),one_one_nat)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5201_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: 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))),N)),one_one_int)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5202_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(aa_real_fun_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_5203_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_5204_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_5205_odd__two__times__div__two__nat,axiom,
    ! [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))
     => ( 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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_5206_power__le__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_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_5207_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_5208_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_5209_even__succ__div__exp,axiom,
    ! [A: code_integer,N: 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),N))
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_5210_even__succ__div__exp,axiom,
    ! [A: nat,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))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( 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))),N)) = 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))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_5211_even__succ__div__exp,axiom,
    ! [A: int,N: 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),N))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_5212_even__succ__mod__exp,axiom,
    ! [A: code_integer,N: 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),N))
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5213_even__succ__mod__exp,axiom,
    ! [A: nat,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))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( 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))),N)) = 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))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5214_even__succ__mod__exp,axiom,
    ! [A: int,N: 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),N))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5215_even__succ__mod__exp,axiom,
    ! [A: code_natural,N: nat] :
      ( dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one)),A)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( modulo8411746178871703098atural(plus_p4538020629002901425atural(one_one_Code_natural,A),power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),N)) = plus_p4538020629002901425atural(one_one_Code_natural,modulo8411746178871703098atural(A,power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5216_sum_Oneutral,axiom,
    ! [A4: set_nat,G2: fun_nat_nat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A4))
         => ( aa_nat_nat(G2,X3) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G2),A4) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_5217_sum_Oneutral,axiom,
    ! [A4: set_int,G2: fun_int_int] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A4))
         => ( aa_int_int(G2,X3) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G2,A4) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_5218_sum_Oneutral,axiom,
    ! [A4: set_nat,G2: fun_nat_real] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A4))
         => ( aa_nat_real(G2,X3) = zero_zero_real ) )
     => ( aa_set_nat_real(groups6591440286371151544t_real(G2),A4) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_5219_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: fun_complex_complex,A4: set_complex] :
      ( ( groups7754918857620584856omplex(G2,A4) != zero_zero_complex )
     => ~ ! [A3: complex] :
            ( pp(aa_set_complex_bool(member_complex(A3),A4))
           => ( aa_complex_complex(G2,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5220_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: fun_real_complex,A4: set_real] :
      ( ( groups5754745047067104278omplex(G2,A4) != zero_zero_complex )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A4))
           => ( aa_real_complex(G2,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5221_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: fun_nat_complex,A4: set_nat] :
      ( ( aa_set_nat_complex(groups2073611262835488442omplex(G2),A4) != zero_zero_complex )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A4))
           => ( aa_nat_complex(G2,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5222_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: fun_int_complex,A4: set_int] :
      ( ( groups3049146728041665814omplex(G2,A4) != zero_zero_complex )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A4))
           => ( aa_int_complex(G2,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5223_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: fun_complex_real,A4: set_complex] :
      ( ( groups5808333547571424918x_real(G2,A4) != zero_zero_real )
     => ~ ! [A3: complex] :
            ( pp(aa_set_complex_bool(member_complex(A3),A4))
           => ( aa_complex_real(G2,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5224_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: fun_real_real,A4: set_real] :
      ( ( groups8097168146408367636l_real(G2,A4) != zero_zero_real )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A4))
           => ( aa_real_real(G2,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5225_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: fun_int_real,A4: set_int] :
      ( ( groups8778361861064173332t_real(G2,A4) != zero_zero_real )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A4))
           => ( aa_int_real(G2,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5226_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: fun_complex_rat,A4: set_complex] :
      ( ( groups5058264527183730370ex_rat(G2,A4) != zero_zero_rat )
     => ~ ! [A3: complex] :
            ( pp(aa_set_complex_bool(member_complex(A3),A4))
           => ( aa_complex_rat(G2,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5227_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: fun_real_rat,A4: set_real] :
      ( ( groups1300246762558778688al_rat(G2,A4) != zero_zero_rat )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A4))
           => ( aa_real_rat(G2,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5228_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: fun_nat_rat,A4: set_nat] :
      ( ( groups2906978787729119204at_rat(G2,A4) != zero_zero_rat )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A4))
           => ( aa_nat_rat(G2,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_5229_sum__mono,axiom,
    ! [K5: set_complex,F2: fun_complex_rat,G2: fun_complex_rat] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F2,I3)),aa_complex_rat(G2,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F2,K5)),groups5058264527183730370ex_rat(G2,K5))) ) ).

% sum_mono
tff(fact_5230_sum__mono,axiom,
    ! [K5: set_real,F2: fun_real_rat,G2: fun_real_rat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F2,I3)),aa_real_rat(G2,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F2,K5)),groups1300246762558778688al_rat(G2,K5))) ) ).

% sum_mono
tff(fact_5231_sum__mono,axiom,
    ! [K5: set_nat,F2: fun_nat_rat,G2: fun_nat_rat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,I3)),aa_nat_rat(G2,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F2,K5)),groups2906978787729119204at_rat(G2,K5))) ) ).

% sum_mono
tff(fact_5232_sum__mono,axiom,
    ! [K5: set_int,F2: fun_int_rat,G2: fun_int_rat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F2,I3)),aa_int_rat(G2,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F2,K5)),groups3906332499630173760nt_rat(G2,K5))) ) ).

% sum_mono
tff(fact_5233_sum__mono,axiom,
    ! [K5: set_complex,F2: fun_complex_nat,G2: fun_complex_nat] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F2,I3)),aa_complex_nat(G2,I3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups5693394587270226106ex_nat(F2,K5)),groups5693394587270226106ex_nat(G2,K5))) ) ).

% sum_mono
tff(fact_5234_sum__mono,axiom,
    ! [K5: set_real,F2: fun_real_nat,G2: fun_real_nat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F2,I3)),aa_real_nat(G2,I3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F2,K5)),groups1935376822645274424al_nat(G2,K5))) ) ).

% sum_mono
tff(fact_5235_sum__mono,axiom,
    ! [K5: set_int,F2: fun_int_nat,G2: fun_int_nat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F2,I3)),aa_int_nat(G2,I3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F2,K5)),groups4541462559716669496nt_nat(G2,K5))) ) ).

% sum_mono
tff(fact_5236_sum__mono,axiom,
    ! [K5: set_complex,F2: fun_complex_int,G2: fun_complex_int] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_complex_int(F2,I3)),aa_complex_int(G2,I3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups5690904116761175830ex_int(F2,K5)),groups5690904116761175830ex_int(G2,K5))) ) ).

% sum_mono
tff(fact_5237_sum__mono,axiom,
    ! [K5: set_real,F2: fun_real_int,G2: fun_real_int] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F2,I3)),aa_real_int(G2,I3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F2,K5)),groups1932886352136224148al_int(G2,K5))) ) ).

% sum_mono
tff(fact_5238_sum__mono,axiom,
    ! [K5: set_nat,F2: fun_nat_int,G2: fun_nat_int] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F2,I3)),aa_nat_int(G2,I3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F2,K5)),groups3539618377306564664at_int(G2,K5))) ) ).

% sum_mono
tff(fact_5239_sum__distrib__left,axiom,
    ! [R: nat,F2: fun_nat_nat,A4: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),aa_set_nat_nat(groups3542108847815614940at_nat(F2),A4)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_eo(R),F2)),A4) ).

% sum_distrib_left
tff(fact_5240_sum__distrib__left,axiom,
    ! [R: int,F2: fun_int_int,A4: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,R),groups4538972089207619220nt_int(F2,A4)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ep(R),F2),A4) ).

% sum_distrib_left
tff(fact_5241_sum__distrib__left,axiom,
    ! [R: real,F2: fun_nat_real,A4: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_set_nat_real(groups6591440286371151544t_real(F2),A4)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_eq(R),F2)),A4) ).

% sum_distrib_left
tff(fact_5242_sum__distrib__right,axiom,
    ! [F2: fun_nat_nat,A4: set_nat,R: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F2),A4)),R) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_er(F2),R)),A4) ).

% sum_distrib_right
tff(fact_5243_sum__distrib__right,axiom,
    ! [F2: fun_int_int,A4: set_int,R: int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F2,A4)),R) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_es(F2),R),A4) ).

% sum_distrib_right
tff(fact_5244_sum__distrib__right,axiom,
    ! [F2: fun_nat_real,A4: set_nat,R: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(F2),A4)),R) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_et(F2),R)),A4) ).

% sum_distrib_right
tff(fact_5245_sum__product,axiom,
    ! [F2: fun_nat_nat,A4: set_nat,G2: fun_nat_nat,B5: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F2),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(G2),B5)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_ev(F2),G2),B5)),A4) ).

% sum_product
tff(fact_5246_sum__product,axiom,
    ! [F2: fun_int_int,A4: set_int,G2: fun_int_int,B5: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F2,A4)),groups4538972089207619220nt_int(G2,B5)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_ex(F2),G2),B5),A4) ).

% sum_product
tff(fact_5247_sum__product,axiom,
    ! [F2: fun_nat_real,A4: set_nat,G2: fun_nat_real,B5: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(F2),A4)),aa_set_nat_real(groups6591440286371151544t_real(G2),B5)) = aa_set_nat_real(groups6591440286371151544t_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_ez(F2),G2),B5)),A4) ).

% sum_product
tff(fact_5248_dvd__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M2))
       => ( M2 = N ) ) ) ).

% dvd_antisym
tff(fact_5249_dvd__trans,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2)) ) ) ).

% dvd_trans
tff(fact_5250_dvd__trans,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2)) ) ) ).

% dvd_trans
tff(fact_5251_dvd__refl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A)) ).

% dvd_refl
tff(fact_5252_dvd__refl,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),A)) ).

% dvd_refl
tff(fact_5253_dvd__field__iff,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B))
    <=> ( ( A = zero_zero_complex )
       => ( B = zero_zero_complex ) ) ) ).

% dvd_field_iff
tff(fact_5254_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_5255_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_5256_dvd__0__left,axiom,
    ! [A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,zero_zero_complex),A))
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
tff(fact_5257_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_5258_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_5259_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_5260_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_5261_dvd__productE,axiom,
    ! [P: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,P),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y3) )
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X3),A))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Y3),B)) ) ) ) ).

% dvd_productE
tff(fact_5262_dvd__productE,axiom,
    ! [P: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,P),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
     => ~ ! [X3: int,Y3: int] :
            ( ( P = aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y3) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X3),A))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Y3),B)) ) ) ) ).

% dvd_productE
tff(fact_5263_division__decomp,axiom,
    ! [A: nat,B: nat,C2: 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),C2)))
     => ? [B9: nat,C5: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B9),C5) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B9),B))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C5),C2)) ) ) ).

% division_decomp
tff(fact_5264_division__decomp,axiom,
    ! [A: int,B: int,C2: 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),C2)))
     => ? [B9: int,C5: int] :
          ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B9),C5) )
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B9),B))
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C5),C2)) ) ) ).

% division_decomp
tff(fact_5265_dvdE,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ~ ! [K: real] : A != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K) ) ).

% dvdE
tff(fact_5266_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ~ ! [K: rat] : A != aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K) ) ).

% dvdE
tff(fact_5267_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ~ ! [K: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) ) ).

% dvdE
tff(fact_5268_dvdE,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ~ ! [K: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B),K) ) ).

% dvdE
tff(fact_5269_dvdI,axiom,
    ! [A: real,B: real,K2: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K2) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A)) ) ).

% dvdI
tff(fact_5270_dvdI,axiom,
    ! [A: rat,B: rat,K2: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K2) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A)) ) ).

% dvdI
tff(fact_5271_dvdI,axiom,
    ! [A: nat,B: nat,K2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% dvdI
tff(fact_5272_dvdI,axiom,
    ! [A: int,B: int,K2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% dvdI
tff(fact_5273_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_5274_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(aa_rat_fun_rat_rat(times_times_rat,B),K3) ) ).

% dvd_def
tff(fact_5275_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_5276_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_5277_dvd__mult,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))) ) ).

% dvd_mult
tff(fact_5278_dvd__mult,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))) ) ).

% dvd_mult
tff(fact_5279_dvd__mult,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2))
     => 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),C2))) ) ).

% dvd_mult
tff(fact_5280_dvd__mult,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2))
     => 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),C2))) ) ).

% dvd_mult
tff(fact_5281_dvd__mult2,axiom,
    ! [A: real,B: real,C2: 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),C2))) ) ).

% dvd_mult2
tff(fact_5282_dvd__mult2,axiom,
    ! [A: rat,B: rat,C2: 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(aa_rat_fun_rat_rat(times_times_rat,B),C2))) ) ).

% dvd_mult2
tff(fact_5283_dvd__mult2,axiom,
    ! [A: nat,B: nat,C2: 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),C2))) ) ).

% dvd_mult2
tff(fact_5284_dvd__mult2,axiom,
    ! [A: int,B: int,C2: 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),C2))) ) ).

% dvd_mult2
tff(fact_5285_dvd__mult__left,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C2)) ) ).

% dvd_mult_left
tff(fact_5286_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C2)) ) ).

% dvd_mult_left
tff(fact_5287_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C2: 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)),C2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2)) ) ).

% dvd_mult_left
tff(fact_5288_dvd__mult__left,axiom,
    ! [A: int,B: int,C2: 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)),C2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2)) ) ).

% dvd_mult_left
tff(fact_5289_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_5290_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ).

% dvd_triv_left
tff(fact_5291_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_5292_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_5293_mult__dvd__mono,axiom,
    ! [A: real,B: real,C2: 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,C2),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_5294_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C2: 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,C2),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_5295_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C2: 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,C2),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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_5296_mult__dvd__mono,axiom,
    ! [A: int,B: int,C2: 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,C2),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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_5297_dvd__mult__right,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),C2)) ) ).

% dvd_mult_right
tff(fact_5298_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),C2)) ) ).

% dvd_mult_right
tff(fact_5299_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C2: 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)),C2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C2)) ) ).

% dvd_mult_right
tff(fact_5300_dvd__mult__right,axiom,
    ! [A: int,B: int,C2: 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)),C2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C2)) ) ).

% dvd_mult_right
tff(fact_5301_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_5302_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A))) ).

% dvd_triv_right
tff(fact_5303_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_5304_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_5305_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C2: 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(plus_plus_real(B),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C2)) ) ) ).

% dvd_add_right_iff
tff(fact_5306_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C2: 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(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C2)) ) ) ).

% dvd_add_right_iff
tff(fact_5307_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C2: 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),C2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2)) ) ) ).

% dvd_add_right_iff
tff(fact_5308_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C2: 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),C2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2)) ) ) ).

% dvd_add_right_iff
tff(fact_5309_dvd__add__left__iff,axiom,
    ! [A: real,C2: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),C2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_5310_dvd__add__left__iff,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_5311_dvd__add__left__iff,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2))
     => ( 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),C2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_5312_dvd__add__left__iff,axiom,
    ! [A: int,C2: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2))
     => ( 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),C2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_5313_dvd__add,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),C2))) ) ) ).

% dvd_add
tff(fact_5314_dvd__add,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2))) ) ) ).

% dvd_add
tff(fact_5315_dvd__add,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
       => 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),C2))) ) ) ).

% dvd_add
tff(fact_5316_dvd__add,axiom,
    ! [A: int,B: int,C2: 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),C2))
       => 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),C2))) ) ) ).

% dvd_add
tff(fact_5317_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer)) ) ) ).

% dvd_unit_imp_unit
tff(fact_5318_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_5319_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_5320_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ).

% unit_imp_dvd
tff(fact_5321_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_5322_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_5323_one__dvd,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,one_one_Code_integer),A)) ).

% one_dvd
tff(fact_5324_one__dvd,axiom,
    ! [A: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,one_one_complex),A)) ).

% one_dvd
tff(fact_5325_one__dvd,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,one_one_real),A)) ).

% one_dvd
tff(fact_5326_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_5327_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_5328_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(aa_rat_fun_rat_rat(minus_minus_rat,Y),Z2))) ) ) ).

% dvd_diff
tff(fact_5329_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_5330_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_5331_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(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),D)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% div_div_div_same
tff(fact_5332_dvd__div__eq__cancel,axiom,
    ! [A: complex,C2: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C2) = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
     => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C2),A))
       => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C2),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_5333_dvd__div__eq__cancel,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),C2) = aa_real_real(divide_divide_real(B),C2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C2),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C2),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_5334_dvd__div__eq__cancel,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C2),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C2),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_5335_dvd__div__eq__cancel,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_5336_dvd__div__eq__cancel,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_5337_dvd__div__eq__iff,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C2),A))
     => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C2),B))
       => ( ( aa_complex_complex(divide1717551699836669952omplex(A),C2) = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_5338_dvd__div__eq__iff,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C2),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C2),B))
       => ( ( aa_real_real(divide_divide_real(A),C2) = aa_real_real(divide_divide_real(B),C2) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_5339_dvd__div__eq__iff,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C2),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C2),B))
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_5340_dvd__div__eq__iff,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_5341_dvd__div__eq__iff,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_5342_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_5343_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_5344_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_5345_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_5346_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_5347_dvd__mod__imp__dvd,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),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,C2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_5348_dvd__mod__imp__dvd,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_5349_dvd__mod__imp__dvd,axiom,
    ! [C2: code_natural,A: code_natural,B: code_natural] :
      ( dvd_dvd_Code_natural(C2,modulo8411746178871703098atural(A,B))
     => ( dvd_dvd_Code_natural(C2,B)
       => dvd_dvd_Code_natural(C2,A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_5350_dvd__mod__iff,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),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,C2),A)) ) ) ).

% dvd_mod_iff
tff(fact_5351_dvd__mod__iff,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),A)) ) ) ).

% dvd_mod_iff
tff(fact_5352_dvd__mod__iff,axiom,
    ! [C2: code_natural,B: code_natural,A: code_natural] :
      ( dvd_dvd_Code_natural(C2,B)
     => ( dvd_dvd_Code_natural(C2,modulo8411746178871703098atural(A,B))
      <=> dvd_dvd_Code_natural(C2,A) ) ) ).

% dvd_mod_iff
tff(fact_5353_sum__nonneg,axiom,
    ! [A4: set_complex,F2: fun_complex_real] :
      ( ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F2,X3))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups5808333547571424918x_real(F2,A4))) ) ).

% sum_nonneg
tff(fact_5354_sum__nonneg,axiom,
    ! [A4: set_real,F2: fun_real_real] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F2,X3))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F2,A4))) ) ).

% sum_nonneg
tff(fact_5355_sum__nonneg,axiom,
    ! [A4: set_int,F2: fun_int_real] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F2,X3))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F2,A4))) ) ).

% sum_nonneg
tff(fact_5356_sum__nonneg,axiom,
    ! [A4: set_complex,F2: fun_complex_rat] :
      ( ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F2,X3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups5058264527183730370ex_rat(F2,A4))) ) ).

% sum_nonneg
tff(fact_5357_sum__nonneg,axiom,
    ! [A4: set_real,F2: fun_real_rat] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F2,X3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F2,A4))) ) ).

% sum_nonneg
tff(fact_5358_sum__nonneg,axiom,
    ! [A4: set_nat,F2: fun_nat_rat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F2,X3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F2,A4))) ) ).

% sum_nonneg
tff(fact_5359_sum__nonneg,axiom,
    ! [A4: set_int,F2: fun_int_rat] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F2,X3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F2,A4))) ) ).

% sum_nonneg
tff(fact_5360_sum__nonneg,axiom,
    ! [A4: set_complex,F2: fun_complex_nat] :
      ( ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F2,X3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups5693394587270226106ex_nat(F2,A4))) ) ).

% sum_nonneg
tff(fact_5361_sum__nonneg,axiom,
    ! [A4: set_real,F2: fun_real_nat] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F2,X3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F2,A4))) ) ).

% sum_nonneg
tff(fact_5362_sum__nonneg,axiom,
    ! [A4: set_int,F2: fun_int_nat] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F2,X3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F2,A4))) ) ).

% sum_nonneg
tff(fact_5363_sum__nonpos,axiom,
    ! [A4: set_complex,F2: fun_complex_real] :
      ( ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F2,X3)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F2,A4)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_5364_sum__nonpos,axiom,
    ! [A4: set_real,F2: fun_real_real] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F2,X3)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F2,A4)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_5365_sum__nonpos,axiom,
    ! [A4: set_int,F2: fun_int_real] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F2,X3)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F2,A4)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_5366_sum__nonpos,axiom,
    ! [A4: set_complex,F2: fun_complex_rat] :
      ( ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F2,X3)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F2,A4)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_5367_sum__nonpos,axiom,
    ! [A4: set_real,F2: fun_real_rat] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F2,X3)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F2,A4)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_5368_sum__nonpos,axiom,
    ! [A4: set_nat,F2: fun_nat_rat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,X3)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F2,A4)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_5369_sum__nonpos,axiom,
    ! [A4: set_int,F2: fun_int_rat] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F2,X3)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F2,A4)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_5370_sum__nonpos,axiom,
    ! [A4: set_complex,F2: fun_complex_nat] :
      ( ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F2,X3)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups5693394587270226106ex_nat(F2,A4)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_5371_sum__nonpos,axiom,
    ! [A4: set_real,F2: fun_real_nat] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F2,X3)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F2,A4)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_5372_sum__nonpos,axiom,
    ! [A4: set_int,F2: fun_int_nat] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F2,X3)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F2,A4)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_5373_dvd__diff__nat,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ) ) ).

% dvd_diff_nat
tff(fact_5374_sum__mono__inv,axiom,
    ! [F2: fun_real_rat,I5: set_real,G2: fun_real_rat,I2: real] :
      ( ( groups1300246762558778688al_rat(F2,I5) = groups1300246762558778688al_rat(G2,I5) )
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F2,I3)),aa_real_rat(G2,I3))) )
       => ( pp(aa_set_real_bool(member_real(I2),I5))
         => ( pp(aa_set_real_bool(finite_finite_real,I5))
           => ( aa_real_rat(F2,I2) = aa_real_rat(G2,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5375_sum__mono__inv,axiom,
    ! [F2: fun_nat_rat,I5: set_nat,G2: fun_nat_rat,I2: nat] :
      ( ( groups2906978787729119204at_rat(F2,I5) = groups2906978787729119204at_rat(G2,I5) )
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,I3)),aa_nat_rat(G2,I3))) )
       => ( pp(aa_set_nat_bool(member_nat(I2),I5))
         => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
           => ( aa_nat_rat(F2,I2) = aa_nat_rat(G2,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5376_sum__mono__inv,axiom,
    ! [F2: fun_int_rat,I5: set_int,G2: fun_int_rat,I2: int] :
      ( ( groups3906332499630173760nt_rat(F2,I5) = groups3906332499630173760nt_rat(G2,I5) )
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F2,I3)),aa_int_rat(G2,I3))) )
       => ( pp(aa_set_int_bool(member_int(I2),I5))
         => ( pp(aa_set_int_bool(finite_finite_int,I5))
           => ( aa_int_rat(F2,I2) = aa_int_rat(G2,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5377_sum__mono__inv,axiom,
    ! [F2: fun_complex_rat,I5: set_complex,G2: fun_complex_rat,I2: complex] :
      ( ( groups5058264527183730370ex_rat(F2,I5) = groups5058264527183730370ex_rat(G2,I5) )
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F2,I3)),aa_complex_rat(G2,I3))) )
       => ( pp(aa_set_complex_bool(member_complex(I2),I5))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
           => ( aa_complex_rat(F2,I2) = aa_complex_rat(G2,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5378_sum__mono__inv,axiom,
    ! [F2: fun_real_nat,I5: set_real,G2: fun_real_nat,I2: real] :
      ( ( groups1935376822645274424al_nat(F2,I5) = groups1935376822645274424al_nat(G2,I5) )
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F2,I3)),aa_real_nat(G2,I3))) )
       => ( pp(aa_set_real_bool(member_real(I2),I5))
         => ( pp(aa_set_real_bool(finite_finite_real,I5))
           => ( aa_real_nat(F2,I2) = aa_real_nat(G2,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5379_sum__mono__inv,axiom,
    ! [F2: fun_int_nat,I5: set_int,G2: fun_int_nat,I2: int] :
      ( ( groups4541462559716669496nt_nat(F2,I5) = groups4541462559716669496nt_nat(G2,I5) )
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F2,I3)),aa_int_nat(G2,I3))) )
       => ( pp(aa_set_int_bool(member_int(I2),I5))
         => ( pp(aa_set_int_bool(finite_finite_int,I5))
           => ( aa_int_nat(F2,I2) = aa_int_nat(G2,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5380_sum__mono__inv,axiom,
    ! [F2: fun_complex_nat,I5: set_complex,G2: fun_complex_nat,I2: complex] :
      ( ( groups5693394587270226106ex_nat(F2,I5) = groups5693394587270226106ex_nat(G2,I5) )
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F2,I3)),aa_complex_nat(G2,I3))) )
       => ( pp(aa_set_complex_bool(member_complex(I2),I5))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
           => ( aa_complex_nat(F2,I2) = aa_complex_nat(G2,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5381_sum__mono__inv,axiom,
    ! [F2: fun_real_int,I5: set_real,G2: fun_real_int,I2: real] :
      ( ( groups1932886352136224148al_int(F2,I5) = groups1932886352136224148al_int(G2,I5) )
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),I5))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F2,I3)),aa_real_int(G2,I3))) )
       => ( pp(aa_set_real_bool(member_real(I2),I5))
         => ( pp(aa_set_real_bool(finite_finite_real,I5))
           => ( aa_real_int(F2,I2) = aa_real_int(G2,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5382_sum__mono__inv,axiom,
    ! [F2: fun_nat_int,I5: set_nat,G2: fun_nat_int,I2: nat] :
      ( ( groups3539618377306564664at_int(F2,I5) = groups3539618377306564664at_int(G2,I5) )
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),I5))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F2,I3)),aa_nat_int(G2,I3))) )
       => ( pp(aa_set_nat_bool(member_nat(I2),I5))
         => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
           => ( aa_nat_int(F2,I2) = aa_nat_int(G2,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5383_sum__mono__inv,axiom,
    ! [F2: fun_complex_int,I5: set_complex,G2: fun_complex_int,I2: complex] :
      ( ( groups5690904116761175830ex_int(F2,I5) = groups5690904116761175830ex_int(G2,I5) )
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),I5))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_complex_int(F2,I3)),aa_complex_int(G2,I3))) )
       => ( pp(aa_set_complex_bool(member_complex(I2),I5))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
           => ( aa_complex_int(F2,I2) = aa_complex_int(G2,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5384_sum__cong__Suc,axiom,
    ! [A4: set_nat,F2: fun_nat_nat,G2: fun_nat_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4))
     => ( ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X3)),A4))
           => ( aa_nat_nat(F2,aa_nat_nat(suc,X3)) = aa_nat_nat(G2,aa_nat_nat(suc,X3)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F2),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(G2),A4) ) ) ) ).

% sum_cong_Suc
tff(fact_5385_sum__cong__Suc,axiom,
    ! [A4: set_nat,F2: fun_nat_real,G2: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4))
     => ( ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X3)),A4))
           => ( aa_nat_real(F2,aa_nat_nat(suc,X3)) = aa_nat_real(G2,aa_nat_nat(suc,X3)) ) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(F2),A4) = aa_set_nat_real(groups6591440286371151544t_real(G2),A4) ) ) ) ).

% sum_cong_Suc
tff(fact_5386_sum_Ointer__filter,axiom,
    ! [A4: set_real,G2: fun_real_complex,P2: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups5754745047067104278omplex(G2,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_fa(A4),P2))) = groups5754745047067104278omplex(aa_fun6813637976755515871omplex(aTP_Lamm_fb(G2),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5387_sum_Ointer__filter,axiom,
    ! [A4: set_nat,G2: fun_nat_complex,P2: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( aa_set_nat_complex(groups2073611262835488442omplex(G2),collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_fc(A4),P2))) = aa_set_nat_complex(groups2073611262835488442omplex(aa_fun8588439219086927967omplex(aTP_Lamm_fd(G2),P2)),A4) ) ) ).

% sum.inter_filter
tff(fact_5388_sum_Ointer__filter,axiom,
    ! [A4: set_int,G2: fun_int_complex,P2: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups3049146728041665814omplex(G2,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_fe(A4),P2))) = groups3049146728041665814omplex(aa_fun3606824691168040799omplex(aTP_Lamm_ff(G2),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5389_sum_Ointer__filter,axiom,
    ! [A4: set_complex,G2: fun_complex_complex,P2: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups7754918857620584856omplex(G2,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_fg(A4),P2))) = groups7754918857620584856omplex(aa_fun1034237174562290143omplex(aTP_Lamm_fh(G2),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5390_sum_Ointer__filter,axiom,
    ! [A4: set_real,G2: fun_real_real,P2: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups8097168146408367636l_real(G2,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_fa(A4),P2))) = groups8097168146408367636l_real(aa_fun6348708079608492509l_real(aTP_Lamm_fi(G2),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5391_sum_Ointer__filter,axiom,
    ! [A4: set_int,G2: fun_int_real,P2: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups8778361861064173332t_real(G2,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_fe(A4),P2))) = groups8778361861064173332t_real(aa_fun6974987029372589917t_real(aTP_Lamm_fj(G2),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5392_sum_Ointer__filter,axiom,
    ! [A4: set_complex,G2: fun_complex_real,P2: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5808333547571424918x_real(G2,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_fg(A4),P2))) = groups5808333547571424918x_real(aa_fun2902006202505214429x_real(aTP_Lamm_fk(G2),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5393_sum_Ointer__filter,axiom,
    ! [A4: set_real,G2: fun_real_rat,P2: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1300246762558778688al_rat(G2,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_fa(A4),P2))) = groups1300246762558778688al_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_fl(G2),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5394_sum_Ointer__filter,axiom,
    ! [A4: set_nat,G2: fun_nat_rat,P2: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( groups2906978787729119204at_rat(G2,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_fc(A4),P2))) = groups2906978787729119204at_rat(aa_fun5656162108953708681at_rat(aTP_Lamm_fm(G2),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5395_sum_Ointer__filter,axiom,
    ! [A4: set_int,G2: fun_int_rat,P2: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups3906332499630173760nt_rat(G2,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_fe(A4),P2))) = groups3906332499630173760nt_rat(aa_fun4400602662665189257nt_rat(aTP_Lamm_fn(G2),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_5396_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_fo,A))),collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_fo,B))))
    <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ).

% subset_divisors_dvd
tff(fact_5397_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_fp,A))),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_fp,B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% subset_divisors_dvd
tff(fact_5398_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_fq,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_fq,B))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% subset_divisors_dvd
tff(fact_5399_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_fr,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_fr,B))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% subset_divisors_dvd
tff(fact_5400_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_set_complex_bool(ord_less_set_complex(collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_fo,A))),collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_fo,B))))
    <=> ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B))
        & ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5401_strict__subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_fp,A))),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_fp,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_5402_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_fr,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_fr,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_5403_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_fq,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_fq,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_5404_sum__le__included,axiom,
    ! [S2: set_int,T: set_int,G2: fun_int_real,I2: fun_int_int,F2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G2,X3))) )
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S2))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_int(I2,Xa) = X3 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F2,X3)),aa_int_real(G2,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F2,S2)),groups8778361861064173332t_real(G2,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5405_sum__le__included,axiom,
    ! [S2: set_int,T: set_complex,G2: fun_complex_real,I2: fun_complex_int,F2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G2,X3))) )
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S2))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_int(I2,Xa) = X3 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F2,X3)),aa_complex_real(G2,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F2,S2)),groups5808333547571424918x_real(G2,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5406_sum__le__included,axiom,
    ! [S2: set_complex,T: set_int,G2: fun_int_real,I2: fun_int_complex,F2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G2,X3))) )
         => ( ! [X3: complex] :
                ( pp(aa_set_complex_bool(member_complex(X3),S2))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_complex(I2,Xa) = X3 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F2,X3)),aa_int_real(G2,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F2,S2)),groups8778361861064173332t_real(G2,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5407_sum__le__included,axiom,
    ! [S2: set_complex,T: set_complex,G2: fun_complex_real,I2: fun_complex_complex,F2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G2,X3))) )
         => ( ! [X3: complex] :
                ( pp(aa_set_complex_bool(member_complex(X3),S2))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_complex(I2,Xa) = X3 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F2,X3)),aa_complex_real(G2,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F2,S2)),groups5808333547571424918x_real(G2,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5408_sum__le__included,axiom,
    ! [S2: set_nat,T: set_nat,G2: fun_nat_rat,I2: fun_nat_nat,F2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T))
       => ( ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G2,X3))) )
         => ( ! [X3: nat] :
                ( pp(aa_set_nat_bool(member_nat(X3),S2))
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_nat(I2,Xa) = X3 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,X3)),aa_nat_rat(G2,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F2,S2)),groups2906978787729119204at_rat(G2,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5409_sum__le__included,axiom,
    ! [S2: set_nat,T: set_int,G2: fun_int_rat,I2: fun_int_nat,F2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G2,X3))) )
         => ( ! [X3: nat] :
                ( pp(aa_set_nat_bool(member_nat(X3),S2))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_nat(I2,Xa) = X3 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,X3)),aa_int_rat(G2,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F2,S2)),groups3906332499630173760nt_rat(G2,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5410_sum__le__included,axiom,
    ! [S2: set_nat,T: set_complex,G2: fun_complex_rat,I2: fun_complex_nat,F2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G2,X3))) )
         => ( ! [X3: nat] :
                ( pp(aa_set_nat_bool(member_nat(X3),S2))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_nat(I2,Xa) = X3 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,X3)),aa_complex_rat(G2,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F2,S2)),groups5058264527183730370ex_rat(G2,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5411_sum__le__included,axiom,
    ! [S2: set_int,T: set_nat,G2: fun_nat_rat,I2: fun_nat_int,F2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T))
       => ( ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G2,X3))) )
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S2))
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_int(I2,Xa) = X3 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F2,X3)),aa_nat_rat(G2,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F2,S2)),groups2906978787729119204at_rat(G2,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5412_sum__le__included,axiom,
    ! [S2: set_int,T: set_int,G2: fun_int_rat,I2: fun_int_int,F2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G2,X3))) )
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S2))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_int(I2,Xa) = X3 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F2,X3)),aa_int_rat(G2,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F2,S2)),groups3906332499630173760nt_rat(G2,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5413_sum__le__included,axiom,
    ! [S2: set_int,T: set_complex,G2: fun_complex_rat,I2: fun_complex_int,F2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G2,X3))) )
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S2))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_int(I2,Xa) = X3 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F2,X3)),aa_complex_rat(G2,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F2,S2)),groups5058264527183730370ex_rat(G2,T))) ) ) ) ) ).

% sum_le_included
tff(fact_5414_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ! [X3: real] :
            ( pp(aa_set_real_bool(member_real(X3),A4))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F2,X3))) )
       => ( ( groups8097168146408367636l_real(F2,A4) = zero_zero_real )
        <=> ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A4))
             => ( aa_real_real(F2,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5415_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A4))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F2,X3))) )
       => ( ( groups8778361861064173332t_real(F2,A4) = zero_zero_real )
        <=> ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A4))
             => ( aa_int_real(F2,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5416_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A4))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F2,X3))) )
       => ( ( groups5808333547571424918x_real(F2,A4) = zero_zero_real )
        <=> ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A4))
             => ( aa_complex_real(F2,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5417_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ! [X3: real] :
            ( pp(aa_set_real_bool(member_real(X3),A4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F2,X3))) )
       => ( ( groups1300246762558778688al_rat(F2,A4) = zero_zero_rat )
        <=> ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A4))
             => ( aa_real_rat(F2,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5418_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_nat,F2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F2,X3))) )
       => ( ( groups2906978787729119204at_rat(F2,A4) = zero_zero_rat )
        <=> ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A4))
             => ( aa_nat_rat(F2,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5419_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F2,X3))) )
       => ( ( groups3906332499630173760nt_rat(F2,A4) = zero_zero_rat )
        <=> ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A4))
             => ( aa_int_rat(F2,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5420_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F2,X3))) )
       => ( ( groups5058264527183730370ex_rat(F2,A4) = zero_zero_rat )
        <=> ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A4))
             => ( aa_complex_rat(F2,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5421_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ! [X3: real] :
            ( pp(aa_set_real_bool(member_real(X3),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F2,X3))) )
       => ( ( groups1935376822645274424al_nat(F2,A4) = zero_zero_nat )
        <=> ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A4))
             => ( aa_real_nat(F2,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5422_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F2,X3))) )
       => ( ( groups4541462559716669496nt_nat(F2,A4) = zero_zero_nat )
        <=> ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A4))
             => ( aa_int_nat(F2,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5423_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F2,X3))) )
       => ( ( groups5693394587270226106ex_nat(F2,A4) = zero_zero_nat )
        <=> ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A4))
             => ( aa_complex_nat(F2,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5424_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F2: fun_int_real,G2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A4))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F2,X3)),aa_int_real(G2,X3))) )
       => ( ? [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),A4))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F2,X5)),aa_int_real(G2,X5))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F2,A4)),groups8778361861064173332t_real(G2,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5425_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F2: fun_complex_real,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A4))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F2,X3)),aa_complex_real(G2,X3))) )
       => ( ? [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),A4))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F2,X5)),aa_complex_real(G2,X5))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F2,A4)),groups5808333547571424918x_real(G2,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5426_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F2: fun_nat_rat,G2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,X3)),aa_nat_rat(G2,X3))) )
       => ( ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),A4))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F2,X5)),aa_nat_rat(G2,X5))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F2,A4)),groups2906978787729119204at_rat(G2,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5427_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F2: fun_int_rat,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F2,X3)),aa_int_rat(G2,X3))) )
       => ( ? [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),A4))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F2,X5)),aa_int_rat(G2,X5))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F2,A4)),groups3906332499630173760nt_rat(G2,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5428_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F2: fun_complex_rat,G2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F2,X3)),aa_complex_rat(G2,X3))) )
       => ( ? [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),A4))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F2,X5)),aa_complex_rat(G2,X5))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F2,A4)),groups5058264527183730370ex_rat(G2,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5429_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F2: fun_int_nat,G2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F2,X3)),aa_int_nat(G2,X3))) )
       => ( ? [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),A4))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F2,X5)),aa_int_nat(G2,X5))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F2,A4)),groups4541462559716669496nt_nat(G2,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5430_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F2: fun_complex_nat,G2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F2,X3)),aa_complex_nat(G2,X3))) )
       => ( ? [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),A4))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F2,X5)),aa_complex_nat(G2,X5))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F2,A4)),groups5693394587270226106ex_nat(G2,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5431_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F2: fun_nat_int,G2: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F2,X3)),aa_nat_int(G2,X3))) )
       => ( ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),A4))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F2,X5)),aa_nat_int(G2,X5))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F2,A4)),groups3539618377306564664at_int(G2,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5432_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F2: fun_complex_int,G2: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_complex_int(F2,X3)),aa_complex_int(G2,X3))) )
       => ( ? [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),A4))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_complex_int(F2,X5)),aa_complex_int(G2,X5))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups5690904116761175830ex_int(F2,A4)),groups5690904116761175830ex_int(G2,A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5433_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F2: fun_nat_nat,G2: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F2,X3)),aa_nat_nat(G2,X3))) )
       => ( ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),A4))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F2,X5)),aa_nat_nat(G2,X5))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F2),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(G2),A4))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5434_sum_Orelated,axiom,
    ! [R2: fun_co1378794544173401837x_bool,S3: set_nat,H: fun_nat_complex,G2: fun_nat_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,zero_zero_complex),zero_zero_complex))
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X15),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,Y15),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_complex_complex(plus_plus_complex(X15),Y15)),aa_complex_complex(plus_plus_complex(X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S3))
         => ( ! [X3: nat] :
                ( pp(aa_set_nat_bool(member_nat(X3),S3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_nat_complex(H,X3)),aa_nat_complex(G2,X3))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_set_nat_complex(groups2073611262835488442omplex(H),S3)),aa_set_nat_complex(groups2073611262835488442omplex(G2),S3))) ) ) ) ) ).

% sum.related
tff(fact_5435_sum_Orelated,axiom,
    ! [R2: fun_co1378794544173401837x_bool,S3: set_int,H: fun_int_complex,G2: fun_int_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,zero_zero_complex),zero_zero_complex))
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X15),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,Y15),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_complex_complex(plus_plus_complex(X15),Y15)),aa_complex_complex(plus_plus_complex(X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_int_complex(H,X3)),aa_int_complex(G2,X3))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,groups3049146728041665814omplex(H,S3)),groups3049146728041665814omplex(G2,S3))) ) ) ) ) ).

% sum.related
tff(fact_5436_sum_Orelated,axiom,
    ! [R2: fun_co1378794544173401837x_bool,S3: set_complex,H: fun_complex_complex,G2: fun_complex_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,zero_zero_complex),zero_zero_complex))
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X15),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,Y15),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_complex_complex(plus_plus_complex(X15),Y15)),aa_complex_complex(plus_plus_complex(X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X3: complex] :
                ( pp(aa_set_complex_bool(member_complex(X3),S3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_complex_complex(H,X3)),aa_complex_complex(G2,X3))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,groups7754918857620584856omplex(H,S3)),groups7754918857620584856omplex(G2,S3))) ) ) ) ) ).

% sum.related
tff(fact_5437_sum_Orelated,axiom,
    ! [R2: fun_re1255401033344146673l_bool,S3: set_int,H: fun_int_real,G2: fun_int_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,zero_zero_real),zero_zero_real))
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,Y15),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_real_real(plus_plus_real(X15),Y15)),aa_real_real(plus_plus_real(X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_int_real(H,X3)),aa_int_real(G2,X3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,groups8778361861064173332t_real(H,S3)),groups8778361861064173332t_real(G2,S3))) ) ) ) ) ).

% sum.related
tff(fact_5438_sum_Orelated,axiom,
    ! [R2: fun_re1255401033344146673l_bool,S3: set_complex,H: fun_complex_real,G2: fun_complex_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,zero_zero_real),zero_zero_real))
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,Y15),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_real_real(plus_plus_real(X15),Y15)),aa_real_real(plus_plus_real(X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X3: complex] :
                ( pp(aa_set_complex_bool(member_complex(X3),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_complex_real(H,X3)),aa_complex_real(G2,X3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,groups5808333547571424918x_real(H,S3)),groups5808333547571424918x_real(G2,S3))) ) ) ) ) ).

% sum.related
tff(fact_5439_sum_Orelated,axiom,
    ! [R2: fun_rat_fun_rat_bool,S3: set_nat,H: fun_nat_rat,G2: fun_nat_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,zero_zero_rat),zero_zero_rat))
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,Y15),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X15),Y15)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S3))
         => ( ! [X3: nat] :
                ( pp(aa_set_nat_bool(member_nat(X3),S3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_nat_rat(H,X3)),aa_nat_rat(G2,X3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,groups2906978787729119204at_rat(H,S3)),groups2906978787729119204at_rat(G2,S3))) ) ) ) ) ).

% sum.related
tff(fact_5440_sum_Orelated,axiom,
    ! [R2: fun_rat_fun_rat_bool,S3: set_int,H: fun_int_rat,G2: fun_int_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,zero_zero_rat),zero_zero_rat))
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,Y15),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X15),Y15)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_int_rat(H,X3)),aa_int_rat(G2,X3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,groups3906332499630173760nt_rat(H,S3)),groups3906332499630173760nt_rat(G2,S3))) ) ) ) ) ).

% sum.related
tff(fact_5441_sum_Orelated,axiom,
    ! [R2: fun_rat_fun_rat_bool,S3: set_complex,H: fun_complex_rat,G2: fun_complex_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,zero_zero_rat),zero_zero_rat))
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,Y15),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X15),Y15)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X3: complex] :
                ( pp(aa_set_complex_bool(member_complex(X3),S3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_complex_rat(H,X3)),aa_complex_rat(G2,X3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,groups5058264527183730370ex_rat(H,S3)),groups5058264527183730370ex_rat(G2,S3))) ) ) ) ) ).

% sum.related
tff(fact_5442_sum_Orelated,axiom,
    ! [R2: fun_nat_fun_nat_bool,S3: set_int,H: fun_int_nat,G2: fun_int_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,zero_zero_nat),zero_zero_nat))
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X15),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,Y15),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,aa_int_nat(H,X3)),aa_int_nat(G2,X3))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,groups4541462559716669496nt_nat(H,S3)),groups4541462559716669496nt_nat(G2,S3))) ) ) ) ) ).

% sum.related
tff(fact_5443_sum_Orelated,axiom,
    ! [R2: fun_nat_fun_nat_bool,S3: set_complex,H: fun_complex_nat,G2: fun_complex_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,zero_zero_nat),zero_zero_nat))
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X15),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,Y15),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X3: complex] :
                ( pp(aa_set_complex_bool(member_complex(X3),S3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,aa_complex_nat(H,X3)),aa_complex_nat(G2,X3))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,groups5693394587270226106ex_nat(H,S3)),groups5693394587270226106ex_nat(G2,S3))) ) ) ) ) ).

% sum.related
tff(fact_5444_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_real,S3: set_real,I2: fun_real_real,J2: fun_real_real,T4: set_real,G2: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T3))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S3,S5)))
             => ( aa_real_real(I2,aa_real_real(J2,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S3,S5)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J2,A3)),minus_minus_set_real(T4,T3))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T4,T3)))
                 => ( aa_real_real(J2,aa_real_real(I2,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T4,T3)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I2,B3)),minus_minus_set_real(S3,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_complex(G2,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T3))
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S3))
                         => ( aa_real_complex(H,aa_real_real(J2,A3)) = aa_real_complex(G2,A3) ) )
                     => ( groups5754745047067104278omplex(G2,S3) = groups5754745047067104278omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5445_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_int,S3: set_real,I2: fun_int_real,J2: fun_real_int,T4: set_int,G2: fun_real_complex,H: fun_int_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S3,S5)))
             => ( aa_int_real(I2,aa_real_int(J2,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S3,S5)))
               => pp(aa_set_int_bool(member_int(aa_real_int(J2,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(J2,aa_int_real(I2,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                   => pp(aa_set_real_bool(member_real(aa_int_real(I2,B3)),minus_minus_set_real(S3,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_complex(G2,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S3))
                         => ( aa_int_complex(H,aa_real_int(J2,A3)) = aa_real_complex(G2,A3) ) )
                     => ( groups5754745047067104278omplex(G2,S3) = groups3049146728041665814omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5446_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_complex,S3: set_real,I2: fun_complex_real,J2: fun_real_complex,T4: set_complex,G2: fun_real_complex,H: fun_complex_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S3,S5)))
             => ( aa_complex_real(I2,aa_real_complex(J2,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S3,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_real_complex(J2,A3)),minus_811609699411566653omplex(T4,T3))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                 => ( aa_real_complex(J2,aa_complex_real(I2,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                   => pp(aa_set_real_bool(member_real(aa_complex_real(I2,B3)),minus_minus_set_real(S3,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_complex(G2,A3) = zero_zero_complex ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T3))
                       => ( aa_complex_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S3))
                         => ( aa_complex_complex(H,aa_real_complex(J2,A3)) = aa_real_complex(G2,A3) ) )
                     => ( groups5754745047067104278omplex(G2,S3) = groups7754918857620584856omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5447_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_real,S3: set_int,I2: fun_real_int,J2: fun_int_real,T4: set_real,G2: fun_int_complex,H: fun_real_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S5)))
             => ( aa_real_int(I2,aa_int_real(J2,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S5)))
               => pp(aa_set_real_bool(member_real(aa_int_real(J2,A3)),minus_minus_set_real(T4,T3))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T4,T3)))
                 => ( aa_int_real(J2,aa_real_int(I2,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T4,T3)))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I2,B3)),minus_minus_set_int(S3,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_complex(G2,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T3))
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S3))
                         => ( aa_real_complex(H,aa_int_real(J2,A3)) = aa_int_complex(G2,A3) ) )
                     => ( groups3049146728041665814omplex(G2,S3) = groups5754745047067104278omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5448_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_int,S3: set_int,I2: fun_int_int,J2: fun_int_int,T4: set_int,G2: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S5)))
             => ( aa_int_int(I2,aa_int_int(J2,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S5)))
               => pp(aa_set_int_bool(member_int(aa_int_int(J2,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(J2,aa_int_int(I2,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(I2,B3)),minus_minus_set_int(S3,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_complex(G2,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S3))
                         => ( aa_int_complex(H,aa_int_int(J2,A3)) = aa_int_complex(G2,A3) ) )
                     => ( groups3049146728041665814omplex(G2,S3) = groups3049146728041665814omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5449_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T3: set_complex,S3: set_int,I2: fun_complex_int,J2: fun_int_complex,T4: set_complex,G2: fun_int_complex,H: fun_complex_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S5)))
             => ( aa_complex_int(I2,aa_int_complex(J2,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_int_complex(J2,A3)),minus_811609699411566653omplex(T4,T3))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                 => ( aa_int_complex(J2,aa_complex_int(I2,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(I2,B3)),minus_minus_set_int(S3,S5))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S5))
                     => ( aa_int_complex(G2,A3) = zero_zero_complex ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T3))
                       => ( aa_complex_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S3))
                         => ( aa_complex_complex(H,aa_int_complex(J2,A3)) = aa_int_complex(G2,A3) ) )
                     => ( groups3049146728041665814omplex(G2,S3) = groups7754918857620584856omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5450_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T3: set_real,S3: set_complex,I2: fun_real_complex,J2: fun_complex_real,T4: set_real,G2: fun_complex_complex,H: fun_real_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T3))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S3,S5)))
             => ( aa_real_complex(I2,aa_complex_real(J2,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S3,S5)))
               => pp(aa_set_real_bool(member_real(aa_complex_real(J2,A3)),minus_minus_set_real(T4,T3))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T4,T3)))
                 => ( aa_complex_real(J2,aa_real_complex(I2,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T4,T3)))
                   => pp(aa_set_complex_bool(member_complex(aa_real_complex(I2,B3)),minus_811609699411566653omplex(S3,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_complex(G2,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T3))
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S3))
                         => ( aa_real_complex(H,aa_complex_real(J2,A3)) = aa_complex_complex(G2,A3) ) )
                     => ( groups7754918857620584856omplex(G2,S3) = groups5754745047067104278omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5451_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T3: set_int,S3: set_complex,I2: fun_int_complex,J2: fun_complex_int,T4: set_int,G2: fun_complex_complex,H: fun_int_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S3,S5)))
             => ( aa_int_complex(I2,aa_complex_int(J2,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S3,S5)))
               => pp(aa_set_int_bool(member_int(aa_complex_int(J2,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(J2,aa_int_complex(I2,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(I2,B3)),minus_811609699411566653omplex(S3,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_complex(G2,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S3))
                         => ( aa_int_complex(H,aa_complex_int(J2,A3)) = aa_complex_complex(G2,A3) ) )
                     => ( groups7754918857620584856omplex(G2,S3) = groups3049146728041665814omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5452_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T3: set_complex,S3: set_complex,I2: fun_complex_complex,J2: fun_complex_complex,T4: set_complex,G2: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S3,S5)))
             => ( aa_complex_complex(I2,aa_complex_complex(J2,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S3,S5)))
               => pp(aa_set_complex_bool(member_complex(aa_complex_complex(J2,A3)),minus_811609699411566653omplex(T4,T3))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                 => ( aa_complex_complex(J2,aa_complex_complex(I2,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                   => pp(aa_set_complex_bool(member_complex(aa_complex_complex(I2,B3)),minus_811609699411566653omplex(S3,S5))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S5))
                     => ( aa_complex_complex(G2,A3) = zero_zero_complex ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T3))
                       => ( aa_complex_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S3))
                         => ( aa_complex_complex(H,aa_complex_complex(J2,A3)) = aa_complex_complex(G2,A3) ) )
                     => ( groups7754918857620584856omplex(G2,S3) = groups7754918857620584856omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5453_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T3: set_real,S3: set_real,I2: fun_real_real,J2: fun_real_real,T4: set_real,G2: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S5))
     => ( pp(aa_set_real_bool(finite_finite_real,T3))
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S3,S5)))
             => ( aa_real_real(I2,aa_real_real(J2,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(S3,S5)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J2,A3)),minus_minus_set_real(T4,T3))) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T4,T3)))
                 => ( aa_real_real(J2,aa_real_real(I2,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(T4,T3)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I2,B3)),minus_minus_set_real(S3,S5))) )
               => ( ! [A3: real] :
                      ( pp(aa_set_real_bool(member_real(A3),S5))
                     => ( aa_real_real(G2,A3) = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( pp(aa_set_real_bool(member_real(B3),T3))
                       => ( aa_real_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( pp(aa_set_real_bool(member_real(A3),S3))
                         => ( aa_real_real(H,aa_real_real(J2,A3)) = aa_real_real(G2,A3) ) )
                     => ( groups8097168146408367636l_real(G2,S3) = groups8097168146408367636l_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5454_not__is__unit__0,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer)) ).

% not_is_unit_0
tff(fact_5455_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_5456_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_5457_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_5458_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(divide_divide_real(A),B) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5459_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(aa_rat_fun_rat_rat(divide_divide_rat,A),B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5460_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_5461_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(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5462_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A) )
      <=> ( B = C2 ) ) ) ).

% unit_mult_right_cancel
tff(fact_5463_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C2: 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,C2),A) )
      <=> ( B = C2 ) ) ) ).

% unit_mult_right_cancel
tff(fact_5464_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C2: 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,C2),A) )
      <=> ( B = C2 ) ) ) ).

% unit_mult_right_cancel
tff(fact_5465_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2) )
      <=> ( B = C2 ) ) ) ).

% unit_mult_left_cancel
tff(fact_5466_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C2: 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),C2) )
      <=> ( B = C2 ) ) ) ).

% unit_mult_left_cancel
tff(fact_5467_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C2: 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),C2) )
      <=> ( B = C2 ) ) ) ).

% unit_mult_left_cancel
tff(fact_5468_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C2))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),C2)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_5469_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C2: 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)),C2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C2)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_5470_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C2: 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)),C2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C2)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_5471_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C2)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_5472_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C2: 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),C2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_5473_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C2: 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),C2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_5474_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C2))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C2)) ) ) ).

% mult_unit_dvd_iff
tff(fact_5475_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C2: 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)),C2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2)) ) ) ).

% mult_unit_dvd_iff
tff(fact_5476_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C2: 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)),C2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2)) ) ) ).

% mult_unit_dvd_iff
tff(fact_5477_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C2)) ) ) ).

% dvd_mult_unit_iff
tff(fact_5478_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C2: 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,C2),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2)) ) ) ).

% dvd_mult_unit_iff
tff(fact_5479_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C2: 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,C2),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2)) ) ) ).

% dvd_mult_unit_iff
tff(fact_5480_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer)) ) ) ).

% is_unit_mult_iff
tff(fact_5481_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_5482_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_5483_dvd__div__mult,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2)),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)),C2) ) ) ).

% dvd_div_mult
tff(fact_5484_dvd__div__mult,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2)),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C2) ) ) ).

% dvd_div_mult
tff(fact_5485_div__mult__swap,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) ) ) ).

% div_mult_swap
tff(fact_5486_div__mult__swap,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2) ) ) ).

% div_mult_swap
tff(fact_5487_div__div__eq__right,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2) ) ) ) ).

% div_div_eq_right
tff(fact_5488_div__div__eq__right,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2) ) ) ) ).

% div_div_eq_right
tff(fact_5489_dvd__div__mult2__eq,axiom,
    ! [B: nat,C2: 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),C2)),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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2) ) ) ).

% dvd_div_mult2_eq
tff(fact_5490_dvd__div__mult2__eq,axiom,
    ! [B: int,C2: 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),C2)),A))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2) ) ) ).

% dvd_div_mult2_eq
tff(fact_5491_dvd__mult__imp__div,axiom,
    ! [A: nat,C2: 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),C2)),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),C2))) ) ).

% dvd_mult_imp_div
tff(fact_5492_dvd__mult__imp__div,axiom,
    ! [A: int,C2: 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),C2)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2))) ) ).

% dvd_mult_imp_div
tff(fact_5493_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C2: 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),C2))
       => ( 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,C2),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_5494_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C2: 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),C2))
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_5495_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(C2,A) )
      <=> ( B = C2 ) ) ) ).

% unit_div_cancel
tff(fact_5496_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C2: 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,C2),A) )
      <=> ( B = C2 ) ) ) ).

% unit_div_cancel
tff(fact_5497_unit__div__cancel,axiom,
    ! [A: int,B: int,C2: 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(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),A) )
      <=> ( B = C2 ) ) ) ).

% unit_div_cancel
tff(fact_5498_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C2))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C2)) ) ) ).

% div_unit_dvd_iff
tff(fact_5499_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C2: 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)),C2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2)) ) ) ).

% div_unit_dvd_iff
tff(fact_5500_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C2: 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(divide_divide_int,A),B)),C2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2)) ) ) ).

% div_unit_dvd_iff
tff(fact_5501_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(C2,B)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C2)) ) ) ).

% dvd_div_unit_iff
tff(fact_5502_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C2: 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,C2),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C2)) ) ) ).

% dvd_div_unit_iff
tff(fact_5503_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C2: 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(divide_divide_int,C2),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C2)) ) ) ).

% dvd_div_unit_iff
tff(fact_5504_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_5505_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_5506_mod__eq__0__iff__dvd,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural(A,B) = zero_z2226904508553997617atural )
    <=> dvd_dvd_Code_natural(B,A) ) ).

% mod_eq_0_iff_dvd
tff(fact_5507_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_5508_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(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_5509_dvd__eq__mod__eq__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( dvd_dvd_Code_natural(A,B)
    <=> ( modulo8411746178871703098atural(B,A) = zero_z2226904508553997617atural ) ) ).

% dvd_eq_mod_eq_0
tff(fact_5510_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_5511_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% mod_0_imp_dvd
tff(fact_5512_mod__0__imp__dvd,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural(A,B) = zero_z2226904508553997617atural )
     => dvd_dvd_Code_natural(B,A) ) ).

% mod_0_imp_dvd
tff(fact_5513_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N: nat,M2: 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,N),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),M2))) ) ) ).

% dvd_power_le
tff(fact_5514_dvd__power__le,axiom,
    ! [X: real,Y: real,N: nat,M2: 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,N),M2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),M2))) ) ) ).

% dvd_power_le
tff(fact_5515_dvd__power__le,axiom,
    ! [X: int,Y: int,N: nat,M2: 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,N),M2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),M2))) ) ) ).

% dvd_power_le
tff(fact_5516_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N: nat,M2: 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,N),M2))
       => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),M2))) ) ) ).

% dvd_power_le
tff(fact_5517_power__le__dvd,axiom,
    ! [A: nat,N: nat,B: nat,M2: 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),N)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),B)) ) ) ).

% power_le_dvd
tff(fact_5518_power__le__dvd,axiom,
    ! [A: real,N: nat,B: real,M2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2)),B)) ) ) ).

% power_le_dvd
tff(fact_5519_power__le__dvd,axiom,
    ! [A: int,N: nat,B: int,M2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),M2)),B)) ) ) ).

% power_le_dvd
tff(fact_5520_power__le__dvd,axiom,
    ! [A: complex,N: nat,B: complex,M2: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),N)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),M2)),B)) ) ) ).

% power_le_dvd
tff(fact_5521_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% le_imp_power_dvd
tff(fact_5522_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% le_imp_power_dvd
tff(fact_5523_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N))) ) ).

% le_imp_power_dvd
tff(fact_5524_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),M2)),aa_nat_complex(power_power_complex(A),N))) ) ).

% le_imp_power_dvd
tff(fact_5525_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_5526_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(aa_int_fun_int_int(modulo_modulo_int,A),B)))) ).

% dvd_minus_mod
tff(fact_5527_dvd__minus__mod,axiom,
    ! [B: code_natural,A: code_natural] : dvd_dvd_Code_natural(B,minus_7197305767214868737atural(A,modulo8411746178871703098atural(A,B))) ).

% dvd_minus_mod
tff(fact_5528_dvd__pos__nat,axiom,
    ! [N: nat,M2: 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(dvd_dvd_nat,M2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2)) ) ) ).

% dvd_pos_nat
tff(fact_5529_nat__dvd__not__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M2)) ) ) ).

% nat_dvd_not_less
tff(fact_5530_pochhammer__pos,axiom,
    ! [X: real,N: 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(aa_real_fun_nat_real(comm_s7457072308508201937r_real,X),N))) ) ).

% pochhammer_pos
tff(fact_5531_pochhammer__pos,axiom,
    ! [X: rat,N: 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),N))) ) ).

% pochhammer_pos
tff(fact_5532_pochhammer__pos,axiom,
    ! [X: nat,N: 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),N))) ) ).

% pochhammer_pos
tff(fact_5533_pochhammer__pos,axiom,
    ! [X: int,N: 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),N))) ) ).

% pochhammer_pos
tff(fact_5534_dvd__minus__self,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N)) ) ) ).

% dvd_minus_self
tff(fact_5535_sum__nonneg__0,axiom,
    ! [S2: set_real,F2: fun_real_real,I2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F2,I3))) )
       => ( ( groups8097168146408367636l_real(F2,S2) = zero_zero_real )
         => ( pp(aa_set_real_bool(member_real(I2),S2))
           => ( aa_real_real(F2,I2) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5536_sum__nonneg__0,axiom,
    ! [S2: set_int,F2: fun_int_real,I2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F2,I3))) )
       => ( ( groups8778361861064173332t_real(F2,S2) = zero_zero_real )
         => ( pp(aa_set_int_bool(member_int(I2),S2))
           => ( aa_int_real(F2,I2) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5537_sum__nonneg__0,axiom,
    ! [S2: set_complex,F2: fun_complex_real,I2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F2,I3))) )
       => ( ( groups5808333547571424918x_real(F2,S2) = zero_zero_real )
         => ( pp(aa_set_complex_bool(member_complex(I2),S2))
           => ( aa_complex_real(F2,I2) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5538_sum__nonneg__0,axiom,
    ! [S2: set_real,F2: fun_real_rat,I2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F2,I3))) )
       => ( ( groups1300246762558778688al_rat(F2,S2) = zero_zero_rat )
         => ( pp(aa_set_real_bool(member_real(I2),S2))
           => ( aa_real_rat(F2,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5539_sum__nonneg__0,axiom,
    ! [S2: set_nat,F2: fun_nat_rat,I2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F2,I3))) )
       => ( ( groups2906978787729119204at_rat(F2,S2) = zero_zero_rat )
         => ( pp(aa_set_nat_bool(member_nat(I2),S2))
           => ( aa_nat_rat(F2,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5540_sum__nonneg__0,axiom,
    ! [S2: set_int,F2: fun_int_rat,I2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F2,I3))) )
       => ( ( groups3906332499630173760nt_rat(F2,S2) = zero_zero_rat )
         => ( pp(aa_set_int_bool(member_int(I2),S2))
           => ( aa_int_rat(F2,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5541_sum__nonneg__0,axiom,
    ! [S2: set_complex,F2: fun_complex_rat,I2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F2,I3))) )
       => ( ( groups5058264527183730370ex_rat(F2,S2) = zero_zero_rat )
         => ( pp(aa_set_complex_bool(member_complex(I2),S2))
           => ( aa_complex_rat(F2,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5542_sum__nonneg__0,axiom,
    ! [S2: set_real,F2: fun_real_nat,I2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F2,I3))) )
       => ( ( groups1935376822645274424al_nat(F2,S2) = zero_zero_nat )
         => ( pp(aa_set_real_bool(member_real(I2),S2))
           => ( aa_real_nat(F2,I2) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5543_sum__nonneg__0,axiom,
    ! [S2: set_int,F2: fun_int_nat,I2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F2,I3))) )
       => ( ( groups4541462559716669496nt_nat(F2,S2) = zero_zero_nat )
         => ( pp(aa_set_int_bool(member_int(I2),S2))
           => ( aa_int_nat(F2,I2) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5544_sum__nonneg__0,axiom,
    ! [S2: set_complex,F2: fun_complex_nat,I2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F2,I3))) )
       => ( ( groups5693394587270226106ex_nat(F2,S2) = zero_zero_nat )
         => ( pp(aa_set_complex_bool(member_complex(I2),S2))
           => ( aa_complex_nat(F2,I2) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5545_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F2: fun_real_real,B5: real,I2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F2,I3))) )
       => ( ( groups8097168146408367636l_real(F2,S2) = B5 )
         => ( pp(aa_set_real_bool(member_real(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F2,I2)),B5)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5546_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F2: fun_int_real,B5: real,I2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F2,I3))) )
       => ( ( groups8778361861064173332t_real(F2,S2) = B5 )
         => ( pp(aa_set_int_bool(member_int(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F2,I2)),B5)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5547_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F2: fun_complex_real,B5: real,I2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F2,I3))) )
       => ( ( groups5808333547571424918x_real(F2,S2) = B5 )
         => ( pp(aa_set_complex_bool(member_complex(I2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F2,I2)),B5)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5548_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F2: fun_real_rat,B5: rat,I2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F2,I3))) )
       => ( ( groups1300246762558778688al_rat(F2,S2) = B5 )
         => ( pp(aa_set_real_bool(member_real(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F2,I2)),B5)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5549_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F2: fun_nat_rat,B5: rat,I2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F2,I3))) )
       => ( ( groups2906978787729119204at_rat(F2,S2) = B5 )
         => ( pp(aa_set_nat_bool(member_nat(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,I2)),B5)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5550_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F2: fun_int_rat,B5: rat,I2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F2,I3))) )
       => ( ( groups3906332499630173760nt_rat(F2,S2) = B5 )
         => ( pp(aa_set_int_bool(member_int(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F2,I2)),B5)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5551_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F2: fun_complex_rat,B5: rat,I2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F2,I3))) )
       => ( ( groups5058264527183730370ex_rat(F2,S2) = B5 )
         => ( pp(aa_set_complex_bool(member_complex(I2),S2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F2,I2)),B5)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5552_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F2: fun_real_nat,B5: nat,I2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F2,I3))) )
       => ( ( groups1935376822645274424al_nat(F2,S2) = B5 )
         => ( pp(aa_set_real_bool(member_real(I2),S2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F2,I2)),B5)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5553_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F2: fun_int_nat,B5: nat,I2: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F2,I3))) )
       => ( ( groups4541462559716669496nt_nat(F2,S2) = B5 )
         => ( pp(aa_set_int_bool(member_int(I2),S2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F2,I2)),B5)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5554_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F2: fun_complex_nat,B5: nat,I2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),S2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F2,I3))) )
       => ( ( groups5693394587270226106ex_nat(F2,S2) = B5 )
         => ( pp(aa_set_complex_bool(member_complex(I2),S2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F2,I2)),B5)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5555_less__eq__dvd__minus,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).

% less_eq_dvd_minus
tff(fact_5556_dvd__diffD1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),M2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),N)) ) ) ) ).

% dvd_diffD1
tff(fact_5557_dvd__diffD,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),M2)) ) ) ) ).

% dvd_diffD
tff(fact_5558_pochhammer__eq__0__mono,axiom,
    ! [A: complex,N: nat,M2: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),M2) = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_5559_pochhammer__eq__0__mono,axiom,
    ! [A: real,N: nat,M2: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),M2) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_5560_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N: nat,M2: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M2) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_5561_pochhammer__neq__0__mono,axiom,
    ! [A: complex,M2: nat,N: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),M2) != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_5562_pochhammer__neq__0__mono,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),M2) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_5563_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M2) != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_5564_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D5: nat,X3: 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),X3) = 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),X3) = 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_5565_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) ) )
         => ? [X3: 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),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),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)),X3) = 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_5566_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D5: nat,X3: 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),X3)),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),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = D5 ) ) ) ).

% bezout1_nat
tff(fact_5567_sum_Ointer__restrict,axiom,
    ! [A4: set_real,G2: fun_real_complex,B5: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups5754745047067104278omplex(G2,inf_inf_set_real(A4,B5)) = groups5754745047067104278omplex(aa_set2225337358267413879omplex(aTP_Lamm_fs(G2),B5),A4) ) ) ).

% sum.inter_restrict
tff(fact_5568_sum_Ointer__restrict,axiom,
    ! [A4: set_int,G2: fun_int_complex,B5: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups3049146728041665814omplex(G2,inf_inf_set_int(A4,B5)) = groups3049146728041665814omplex(aa_set2721330520047917815omplex(aTP_Lamm_ft(G2),B5),A4) ) ) ).

% sum.inter_restrict
tff(fact_5569_sum_Ointer__restrict,axiom,
    ! [A4: set_complex,G2: fun_complex_complex,B5: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups7754918857620584856omplex(G2,inf_inf_set_complex(A4,B5)) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aTP_Lamm_fu(G2),B5),A4) ) ) ).

% sum.inter_restrict
tff(fact_5570_sum_Ointer__restrict,axiom,
    ! [A4: set_real,G2: fun_real_real,B5: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups8097168146408367636l_real(G2,inf_inf_set_real(A4,B5)) = groups8097168146408367636l_real(aa_set8744029839670122869l_real(aTP_Lamm_fv(G2),B5),A4) ) ) ).

% sum.inter_restrict
tff(fact_5571_sum_Ointer__restrict,axiom,
    ! [A4: set_int,G2: fun_int_real,B5: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups8778361861064173332t_real(G2,inf_inf_set_int(A4,B5)) = groups8778361861064173332t_real(aa_set6366552415685541621t_real(aTP_Lamm_fw(G2),B5),A4) ) ) ).

% sum.inter_restrict
tff(fact_5572_sum_Ointer__restrict,axiom,
    ! [A4: set_complex,G2: fun_complex_real,B5: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5808333547571424918x_real(G2,inf_inf_set_complex(A4,B5)) = groups5808333547571424918x_real(aa_set904117813986905849x_real(aTP_Lamm_fx(G2),B5),A4) ) ) ).

% sum.inter_restrict
tff(fact_5573_sum_Ointer__restrict,axiom,
    ! [A4: set_real,G2: fun_real_rat,B5: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1300246762558778688al_rat(G2,inf_inf_set_real(A4,B5)) = groups1300246762558778688al_rat(aa_set4110029769364456865al_rat(aTP_Lamm_fy(G2),B5),A4) ) ) ).

% sum.inter_restrict
tff(fact_5574_sum_Ointer__restrict,axiom,
    ! [A4: set_int,G2: fun_int_rat,B5: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups3906332499630173760nt_rat(G2,inf_inf_set_int(A4,B5)) = groups3906332499630173760nt_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_fz(G2),B5),A4) ) ) ).

% sum.inter_restrict
tff(fact_5575_sum_Ointer__restrict,axiom,
    ! [A4: set_complex,G2: fun_complex_rat,B5: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5058264527183730370ex_rat(G2,inf_inf_set_complex(A4,B5)) = groups5058264527183730370ex_rat(aa_set7482584248617139493ex_rat(aTP_Lamm_ga(G2),B5),A4) ) ) ).

% sum.inter_restrict
tff(fact_5576_sum_Ointer__restrict,axiom,
    ! [A4: set_real,G2: fun_real_nat,B5: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1935376822645274424al_nat(G2,inf_inf_set_real(A4,B5)) = groups1935376822645274424al_nat(aa_set7618258327757853593al_nat(aTP_Lamm_gb(G2),B5),A4) ) ) ).

% sum.inter_restrict
tff(fact_5577_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G2: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups5754745047067104278omplex(G2,minus_minus_set_real(A4,collect_real(aTP_Lamm_gc(G2)))) = groups5754745047067104278omplex(G2,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5578_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G2: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups3049146728041665814omplex(G2,minus_minus_set_int(A4,collect_int(aTP_Lamm_gd(G2)))) = groups3049146728041665814omplex(G2,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5579_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G2: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups7754918857620584856omplex(G2,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_ge(G2)))) = groups7754918857620584856omplex(G2,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5580_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups8097168146408367636l_real(G2,minus_minus_set_real(A4,collect_real(aTP_Lamm_gf(G2)))) = groups8097168146408367636l_real(G2,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5581_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups8778361861064173332t_real(G2,minus_minus_set_int(A4,collect_int(aTP_Lamm_gg(G2)))) = groups8778361861064173332t_real(G2,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5582_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5808333547571424918x_real(G2,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_gh(G2)))) = groups5808333547571424918x_real(G2,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5583_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1300246762558778688al_rat(G2,minus_minus_set_real(A4,collect_real(aTP_Lamm_gi(G2)))) = groups1300246762558778688al_rat(G2,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5584_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups3906332499630173760nt_rat(G2,minus_minus_set_int(A4,collect_int(aTP_Lamm_gj(G2)))) = groups3906332499630173760nt_rat(G2,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5585_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5058264527183730370ex_rat(G2,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_gk(G2)))) = groups5058264527183730370ex_rat(G2,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5586_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1935376822645274424al_nat(G2,minus_minus_set_real(A4,collect_real(aTP_Lamm_gl(G2)))) = groups1935376822645274424al_nat(G2,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5587_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G2: fun_nat_nat,M2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gm(G2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_5588_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G2: fun_nat_real,M2: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),aa_nat_nat(suc,N))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gn(G2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_5589_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G2: fun_nat_nat,M2: nat,K2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K2))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_go(G2),K2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_5590_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G2: fun_nat_real,M2: nat,K2: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K2))) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_gp(G2),K2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_5591_sum__pos2,axiom,
    ! [I5: set_real,I2: real,F2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( pp(aa_set_real_bool(member_real(I2),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F2,I2)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F2,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F2,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_5592_sum__pos2,axiom,
    ! [I5: set_int,I2: int,F2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( 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(F2,I2)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F2,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F2,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_5593_sum__pos2,axiom,
    ! [I5: set_complex,I2: complex,F2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( 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(F2,I2)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F2,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F2,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_5594_sum__pos2,axiom,
    ! [I5: set_real,I2: real,F2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( pp(aa_set_real_bool(member_real(I2),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F2,I2)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F2,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F2,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_5595_sum__pos2,axiom,
    ! [I5: set_nat,I2: nat,F2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( 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(F2,I2)))
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F2,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F2,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_5596_sum__pos2,axiom,
    ! [I5: set_int,I2: int,F2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( 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(F2,I2)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F2,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F2,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_5597_sum__pos2,axiom,
    ! [I5: set_complex,I2: complex,F2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( 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(F2,I2)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F2,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F2,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_5598_sum__pos2,axiom,
    ! [I5: set_real,I2: real,F2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( pp(aa_set_real_bool(member_real(I2),I5))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F2,I2)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I5))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F2,I3))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F2,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_5599_sum__pos2,axiom,
    ! [I5: set_int,I2: int,F2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( pp(aa_set_int_bool(member_int(I2),I5))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F2,I2)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I5))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F2,I3))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F2,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_5600_sum__pos2,axiom,
    ! [I5: set_complex,I2: complex,F2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( 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(F2,I2)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I5))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F2,I3))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F2,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_5601_sum__pos,axiom,
    ! [I5: set_complex,F2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F2,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F2,I5))) ) ) ) ).

% sum_pos
tff(fact_5602_sum__pos,axiom,
    ! [I5: set_int,F2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F2,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F2,I5))) ) ) ) ).

% sum_pos
tff(fact_5603_sum__pos,axiom,
    ! [I5: set_real,F2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F2,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F2,I5))) ) ) ) ).

% sum_pos
tff(fact_5604_sum__pos,axiom,
    ! [I5: set_complex,F2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F2,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F2,I5))) ) ) ) ).

% sum_pos
tff(fact_5605_sum__pos,axiom,
    ! [I5: set_nat,F2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( pp(aa_set_nat_bool(member_nat(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F2,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F2,I5))) ) ) ) ).

% sum_pos
tff(fact_5606_sum__pos,axiom,
    ! [I5: set_int,F2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F2,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F2,I5))) ) ) ) ).

% sum_pos
tff(fact_5607_sum__pos,axiom,
    ! [I5: set_real,F2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F2,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F2,I5))) ) ) ) ).

% sum_pos
tff(fact_5608_sum__pos,axiom,
    ! [I5: set_complex,F2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I5))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F2,I3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F2,I5))) ) ) ) ).

% sum_pos
tff(fact_5609_sum__pos,axiom,
    ! [I5: set_int,F2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I5))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F2,I3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F2,I5))) ) ) ) ).

% sum_pos
tff(fact_5610_sum__pos,axiom,
    ! [I5: set_real,F2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I5))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F2,I3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F2,I5))) ) ) ) ).

% sum_pos
tff(fact_5611_sum__bounded__below,axiom,
    ! [A4: set_real,K5: real,F2: fun_real_real] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K5),aa_real_real(F2,I3))) )
     => 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,finite_card_real(A4))),K5)),groups8097168146408367636l_real(F2,A4))) ) ).

% sum_bounded_below
tff(fact_5612_sum__bounded__below,axiom,
    ! [A4: set_complex,K5: real,F2: fun_complex_real] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K5),aa_complex_real(F2,I3))) )
     => 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,finite_card_complex(A4))),K5)),groups5808333547571424918x_real(F2,A4))) ) ).

% sum_bounded_below
tff(fact_5613_sum__bounded__below,axiom,
    ! [A4: set_int,K5: real,F2: fun_int_real] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K5),aa_int_real(F2,I3))) )
     => 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,finite_card_int(A4))),K5)),groups8778361861064173332t_real(F2,A4))) ) ).

% sum_bounded_below
tff(fact_5614_sum__bounded__below,axiom,
    ! [A4: set_real,K5: rat,F2: fun_real_rat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K5),aa_real_rat(F2,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_real(A4))),K5)),groups1300246762558778688al_rat(F2,A4))) ) ).

% sum_bounded_below
tff(fact_5615_sum__bounded__below,axiom,
    ! [A4: set_nat,K5: rat,F2: fun_nat_rat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K5),aa_nat_rat(F2,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A4))),K5)),groups2906978787729119204at_rat(F2,A4))) ) ).

% sum_bounded_below
tff(fact_5616_sum__bounded__below,axiom,
    ! [A4: set_complex,K5: rat,F2: fun_complex_rat] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K5),aa_complex_rat(F2,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_complex(A4))),K5)),groups5058264527183730370ex_rat(F2,A4))) ) ).

% sum_bounded_below
tff(fact_5617_sum__bounded__below,axiom,
    ! [A4: set_int,K5: rat,F2: fun_int_rat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K5),aa_int_rat(F2,I3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_int(A4))),K5)),groups3906332499630173760nt_rat(F2,A4))) ) ).

% sum_bounded_below
tff(fact_5618_sum__bounded__below,axiom,
    ! [A4: set_real,K5: nat,F2: fun_real_nat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K5),aa_real_nat(F2,I3))) )
     => 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(semiri1316708129612266289at_nat,finite_card_real(A4))),K5)),groups1935376822645274424al_nat(F2,A4))) ) ).

% sum_bounded_below
tff(fact_5619_sum__bounded__below,axiom,
    ! [A4: set_complex,K5: nat,F2: fun_complex_nat] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K5),aa_complex_nat(F2,I3))) )
     => 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(semiri1316708129612266289at_nat,finite_card_complex(A4))),K5)),groups5693394587270226106ex_nat(F2,A4))) ) ).

% sum_bounded_below
tff(fact_5620_sum__bounded__below,axiom,
    ! [A4: set_int,K5: nat,F2: fun_int_nat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K5),aa_int_nat(F2,I3))) )
     => 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(semiri1316708129612266289at_nat,finite_card_int(A4))),K5)),groups4541462559716669496nt_nat(F2,A4))) ) ).

% sum_bounded_below
tff(fact_5621_sum__bounded__above,axiom,
    ! [A4: set_real,F2: fun_real_real,K5: real] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F2,I3)),K5)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F2,A4)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_real(A4))),K5))) ) ).

% sum_bounded_above
tff(fact_5622_sum__bounded__above,axiom,
    ! [A4: set_complex,F2: fun_complex_real,K5: real] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F2,I3)),K5)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F2,A4)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_complex(A4))),K5))) ) ).

% sum_bounded_above
tff(fact_5623_sum__bounded__above,axiom,
    ! [A4: set_int,F2: fun_int_real,K5: real] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F2,I3)),K5)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F2,A4)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_int(A4))),K5))) ) ).

% sum_bounded_above
tff(fact_5624_sum__bounded__above,axiom,
    ! [A4: set_real,F2: fun_real_rat,K5: rat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F2,I3)),K5)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F2,A4)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_real(A4))),K5))) ) ).

% sum_bounded_above
tff(fact_5625_sum__bounded__above,axiom,
    ! [A4: set_nat,F2: fun_nat_rat,K5: rat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,I3)),K5)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F2,A4)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A4))),K5))) ) ).

% sum_bounded_above
tff(fact_5626_sum__bounded__above,axiom,
    ! [A4: set_complex,F2: fun_complex_rat,K5: rat] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F2,I3)),K5)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F2,A4)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_complex(A4))),K5))) ) ).

% sum_bounded_above
tff(fact_5627_sum__bounded__above,axiom,
    ! [A4: set_int,F2: fun_int_rat,K5: rat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F2,I3)),K5)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F2,A4)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_int(A4))),K5))) ) ).

% sum_bounded_above
tff(fact_5628_sum__bounded__above,axiom,
    ! [A4: set_real,F2: fun_real_nat,K5: nat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F2,I3)),K5)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F2,A4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_real(A4))),K5))) ) ).

% sum_bounded_above
tff(fact_5629_sum__bounded__above,axiom,
    ! [A4: set_complex,F2: fun_complex_nat,K5: nat] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F2,I3)),K5)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups5693394587270226106ex_nat(F2,A4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_complex(A4))),K5))) ) ).

% sum_bounded_above
tff(fact_5630_sum__bounded__above,axiom,
    ! [A4: set_int,F2: fun_int_nat,K5: nat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F2,I3)),K5)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F2,A4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_int(A4))),K5))) ) ).

% sum_bounded_above
tff(fact_5631_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G2: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T4))
       => ( ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(T4,S3)))
             => ( aa_real_complex(G2,X3) = zero_zero_complex ) )
         => ( ! [X3: real] :
                ( pp(aa_set_real_bool(member_real(X3),S3))
               => ( aa_real_complex(G2,X3) = aa_real_complex(H,X3) ) )
           => ( groups5754745047067104278omplex(G2,T4) = groups5754745047067104278omplex(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5632_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_int,S3: set_int,G2: 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))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T4,S3)))
             => ( aa_int_complex(G2,X3) = zero_zero_complex ) )
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S3))
               => ( aa_int_complex(G2,X3) = aa_int_complex(H,X3) ) )
           => ( groups3049146728041665814omplex(G2,T4) = groups3049146728041665814omplex(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5633_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S3: set_complex,G2: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_complex(G2,X3) = zero_zero_complex ) )
         => ( ! [X3: complex] :
                ( pp(aa_set_complex_bool(member_complex(X3),S3))
               => ( aa_complex_complex(G2,X3) = aa_complex_complex(H,X3) ) )
           => ( groups7754918857620584856omplex(G2,T4) = groups7754918857620584856omplex(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5634_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G2: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T4))
       => ( ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(T4,S3)))
             => ( aa_real_real(G2,X3) = zero_zero_real ) )
         => ( ! [X3: real] :
                ( pp(aa_set_real_bool(member_real(X3),S3))
               => ( aa_real_real(G2,X3) = aa_real_real(H,X3) ) )
           => ( groups8097168146408367636l_real(G2,T4) = groups8097168146408367636l_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5635_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_int,S3: set_int,G2: 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))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T4,S3)))
             => ( aa_int_real(G2,X3) = zero_zero_real ) )
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S3))
               => ( aa_int_real(G2,X3) = aa_int_real(H,X3) ) )
           => ( groups8778361861064173332t_real(G2,T4) = groups8778361861064173332t_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5636_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S3: set_complex,G2: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_real(G2,X3) = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( pp(aa_set_complex_bool(member_complex(X3),S3))
               => ( aa_complex_real(G2,X3) = aa_complex_real(H,X3) ) )
           => ( groups5808333547571424918x_real(G2,T4) = groups5808333547571424918x_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5637_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G2: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T4))
       => ( ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(T4,S3)))
             => ( aa_real_rat(G2,X3) = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( pp(aa_set_real_bool(member_real(X3),S3))
               => ( aa_real_rat(G2,X3) = aa_real_rat(H,X3) ) )
           => ( groups1300246762558778688al_rat(G2,T4) = groups1300246762558778688al_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5638_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_int,S3: set_int,G2: 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))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T4,S3)))
             => ( aa_int_rat(G2,X3) = zero_zero_rat ) )
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S3))
               => ( aa_int_rat(G2,X3) = aa_int_rat(H,X3) ) )
           => ( groups3906332499630173760nt_rat(G2,T4) = groups3906332499630173760nt_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5639_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S3: set_complex,G2: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_rat(G2,X3) = zero_zero_rat ) )
         => ( ! [X3: complex] :
                ( pp(aa_set_complex_bool(member_complex(X3),S3))
               => ( aa_complex_rat(G2,X3) = aa_complex_rat(H,X3) ) )
           => ( groups5058264527183730370ex_rat(G2,T4) = groups5058264527183730370ex_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5640_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G2: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T4))
       => ( ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(T4,S3)))
             => ( aa_real_nat(G2,X3) = zero_zero_nat ) )
         => ( ! [X3: real] :
                ( pp(aa_set_real_bool(member_real(X3),S3))
               => ( aa_real_nat(G2,X3) = aa_real_nat(H,X3) ) )
           => ( groups1935376822645274424al_nat(G2,T4) = groups1935376822645274424al_nat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5641_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H: fun_real_complex,G2: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T4))
       => ( ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(T4,S3)))
             => ( aa_real_complex(H,X3) = zero_zero_complex ) )
         => ( ! [X3: real] :
                ( pp(aa_set_real_bool(member_real(X3),S3))
               => ( aa_real_complex(G2,X3) = aa_real_complex(H,X3) ) )
           => ( groups5754745047067104278omplex(G2,S3) = groups5754745047067104278omplex(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5642_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_int,S3: set_int,H: fun_int_complex,G2: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T4,S3)))
             => ( aa_int_complex(H,X3) = zero_zero_complex ) )
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S3))
               => ( aa_int_complex(G2,X3) = aa_int_complex(H,X3) ) )
           => ( groups3049146728041665814omplex(G2,S3) = groups3049146728041665814omplex(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5643_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S3: set_complex,H: fun_complex_complex,G2: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_complex(H,X3) = zero_zero_complex ) )
         => ( ! [X3: complex] :
                ( pp(aa_set_complex_bool(member_complex(X3),S3))
               => ( aa_complex_complex(G2,X3) = aa_complex_complex(H,X3) ) )
           => ( groups7754918857620584856omplex(G2,S3) = groups7754918857620584856omplex(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5644_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H: fun_real_real,G2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T4))
       => ( ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(T4,S3)))
             => ( aa_real_real(H,X3) = zero_zero_real ) )
         => ( ! [X3: real] :
                ( pp(aa_set_real_bool(member_real(X3),S3))
               => ( aa_real_real(G2,X3) = aa_real_real(H,X3) ) )
           => ( groups8097168146408367636l_real(G2,S3) = groups8097168146408367636l_real(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5645_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_int,S3: set_int,H: fun_int_real,G2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T4,S3)))
             => ( aa_int_real(H,X3) = zero_zero_real ) )
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S3))
               => ( aa_int_real(G2,X3) = aa_int_real(H,X3) ) )
           => ( groups8778361861064173332t_real(G2,S3) = groups8778361861064173332t_real(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5646_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S3: set_complex,H: fun_complex_real,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_real(H,X3) = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( pp(aa_set_complex_bool(member_complex(X3),S3))
               => ( aa_complex_real(G2,X3) = aa_complex_real(H,X3) ) )
           => ( groups5808333547571424918x_real(G2,S3) = groups5808333547571424918x_real(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5647_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H: fun_real_rat,G2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T4))
       => ( ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(T4,S3)))
             => ( aa_real_rat(H,X3) = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( pp(aa_set_real_bool(member_real(X3),S3))
               => ( aa_real_rat(G2,X3) = aa_real_rat(H,X3) ) )
           => ( groups1300246762558778688al_rat(G2,S3) = groups1300246762558778688al_rat(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5648_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_int,S3: set_int,H: fun_int_rat,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T4,S3)))
             => ( aa_int_rat(H,X3) = zero_zero_rat ) )
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S3))
               => ( aa_int_rat(G2,X3) = aa_int_rat(H,X3) ) )
           => ( groups3906332499630173760nt_rat(G2,S3) = groups3906332499630173760nt_rat(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5649_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S3: set_complex,H: fun_complex_rat,G2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_rat(H,X3) = zero_zero_rat ) )
         => ( ! [X3: complex] :
                ( pp(aa_set_complex_bool(member_complex(X3),S3))
               => ( aa_complex_rat(G2,X3) = aa_complex_rat(H,X3) ) )
           => ( groups5058264527183730370ex_rat(G2,S3) = groups5058264527183730370ex_rat(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5650_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H: fun_real_nat,G2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T4))
       => ( ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(T4,S3)))
             => ( aa_real_nat(H,X3) = zero_zero_nat ) )
         => ( ! [X3: real] :
                ( pp(aa_set_real_bool(member_real(X3),S3))
               => ( aa_real_nat(G2,X3) = aa_real_nat(H,X3) ) )
           => ( groups1935376822645274424al_nat(G2,S3) = groups1935376822645274424al_nat(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5651_sum_Omono__neutral__right,axiom,
    ! [T4: set_int,S3: set_int,G2: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T4,S3)))
             => ( aa_int_complex(G2,X3) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G2,T4) = groups3049146728041665814omplex(G2,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5652_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G2: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_complex(G2,X3) = zero_zero_complex ) )
         => ( groups7754918857620584856omplex(G2,T4) = groups7754918857620584856omplex(G2,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5653_sum_Omono__neutral__right,axiom,
    ! [T4: set_int,S3: set_int,G2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T4,S3)))
             => ( aa_int_real(G2,X3) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G2,T4) = groups8778361861064173332t_real(G2,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5654_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_real(G2,X3) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G2,T4) = groups5808333547571424918x_real(G2,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5655_sum_Omono__neutral__right,axiom,
    ! [T4: set_int,S3: set_int,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T4,S3)))
             => ( aa_int_rat(G2,X3) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G2,T4) = groups3906332499630173760nt_rat(G2,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5656_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_rat(G2,X3) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G2,T4) = groups5058264527183730370ex_rat(G2,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5657_sum_Omono__neutral__right,axiom,
    ! [T4: set_int,S3: set_int,G2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T4,S3)))
             => ( aa_int_nat(G2,X3) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G2,T4) = groups4541462559716669496nt_nat(G2,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5658_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_nat(G2,X3) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G2,T4) = groups5693394587270226106ex_nat(G2,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5659_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G2: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_int(G2,X3) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G2,T4) = groups5690904116761175830ex_int(G2,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5660_sum_Omono__neutral__right,axiom,
    ! [T4: set_nat,S3: set_nat,G2: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T4))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(S3),T4))
       => ( ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T4),S3)))
             => ( aa_nat_complex(G2,X3) = zero_zero_complex ) )
         => ( aa_set_nat_complex(groups2073611262835488442omplex(G2),T4) = aa_set_nat_complex(groups2073611262835488442omplex(G2),S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5661_sum_Omono__neutral__left,axiom,
    ! [T4: set_int,S3: set_int,G2: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T4,S3)))
             => ( aa_int_complex(G2,X3) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G2,S3) = groups3049146728041665814omplex(G2,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5662_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G2: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_complex(G2,X3) = zero_zero_complex ) )
         => ( groups7754918857620584856omplex(G2,S3) = groups7754918857620584856omplex(G2,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5663_sum_Omono__neutral__left,axiom,
    ! [T4: set_int,S3: set_int,G2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T4,S3)))
             => ( aa_int_real(G2,X3) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G2,S3) = groups8778361861064173332t_real(G2,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5664_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_real(G2,X3) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G2,S3) = groups5808333547571424918x_real(G2,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5665_sum_Omono__neutral__left,axiom,
    ! [T4: set_int,S3: set_int,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T4,S3)))
             => ( aa_int_rat(G2,X3) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G2,S3) = groups3906332499630173760nt_rat(G2,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5666_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_rat(G2,X3) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G2,S3) = groups5058264527183730370ex_rat(G2,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5667_sum_Omono__neutral__left,axiom,
    ! [T4: set_int,S3: set_int,G2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T4,S3)))
             => ( aa_int_nat(G2,X3) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G2,S3) = groups4541462559716669496nt_nat(G2,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5668_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_nat(G2,X3) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G2,S3) = groups5693394587270226106ex_nat(G2,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5669_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G2: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_int(G2,X3) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G2,S3) = groups5690904116761175830ex_int(G2,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5670_sum_Omono__neutral__left,axiom,
    ! [T4: set_nat,S3: set_nat,G2: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,T4))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(S3),T4))
       => ( ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T4),S3)))
             => ( aa_nat_complex(G2,X3) = zero_zero_complex ) )
         => ( aa_set_nat_complex(groups2073611262835488442omplex(G2),S3) = aa_set_nat_complex(groups2073611262835488442omplex(G2),T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5671_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G2: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B5),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_complex(G2,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B5)))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G2,C4) = groups5754745047067104278omplex(H,C4) )
               => ( groups5754745047067104278omplex(G2,A4) = groups5754745047067104278omplex(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5672_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B5: set_int,G2: 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(A4),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B5),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A4)))
               => ( aa_int_complex(G2,A3) = zero_zero_complex ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B5)))
                 => ( aa_int_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups3049146728041665814omplex(G2,C4) = groups3049146728041665814omplex(H,C4) )
               => ( groups3049146728041665814omplex(G2,A4) = groups3049146728041665814omplex(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5673_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G2: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B5),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_complex(G2,A3) = zero_zero_complex ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B5)))
                 => ( aa_complex_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups7754918857620584856omplex(G2,C4) = groups7754918857620584856omplex(H,C4) )
               => ( groups7754918857620584856omplex(G2,A4) = groups7754918857620584856omplex(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5674_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G2: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B5),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_real(G2,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B5)))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G2,C4) = groups8097168146408367636l_real(H,C4) )
               => ( groups8097168146408367636l_real(G2,A4) = groups8097168146408367636l_real(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5675_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B5: set_int,G2: 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(A4),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B5),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A4)))
               => ( aa_int_real(G2,A3) = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B5)))
                 => ( aa_int_real(H,B3) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G2,C4) = groups8778361861064173332t_real(H,C4) )
               => ( groups8778361861064173332t_real(G2,A4) = groups8778361861064173332t_real(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5676_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G2: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B5),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_real(G2,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B5)))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G2,C4) = groups5808333547571424918x_real(H,C4) )
               => ( groups5808333547571424918x_real(G2,A4) = groups5808333547571424918x_real(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5677_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G2: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B5),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_rat(G2,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B5)))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G2,C4) = groups1300246762558778688al_rat(H,C4) )
               => ( groups1300246762558778688al_rat(G2,A4) = groups1300246762558778688al_rat(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5678_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B5: set_int,G2: 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(A4),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B5),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A4)))
               => ( aa_int_rat(G2,A3) = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B5)))
                 => ( aa_int_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G2,C4) = groups3906332499630173760nt_rat(H,C4) )
               => ( groups3906332499630173760nt_rat(G2,A4) = groups3906332499630173760nt_rat(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5679_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G2: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B5),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_rat(G2,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B5)))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G2,C4) = groups5058264527183730370ex_rat(H,C4) )
               => ( groups5058264527183730370ex_rat(G2,A4) = groups5058264527183730370ex_rat(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5680_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G2: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B5),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_nat(G2,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B5)))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G2,C4) = groups1935376822645274424al_nat(H,C4) )
               => ( groups1935376822645274424al_nat(G2,A4) = groups1935376822645274424al_nat(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5681_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G2: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B5),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_complex(G2,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B5)))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G2,A4) = groups5754745047067104278omplex(H,B5) )
              <=> ( groups5754745047067104278omplex(G2,C4) = groups5754745047067104278omplex(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5682_sum_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B5: set_int,G2: 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(A4),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B5),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A4)))
               => ( aa_int_complex(G2,A3) = zero_zero_complex ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B5)))
                 => ( aa_int_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups3049146728041665814omplex(G2,A4) = groups3049146728041665814omplex(H,B5) )
              <=> ( groups3049146728041665814omplex(G2,C4) = groups3049146728041665814omplex(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5683_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G2: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B5),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_complex(G2,A3) = zero_zero_complex ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B5)))
                 => ( aa_complex_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups7754918857620584856omplex(G2,A4) = groups7754918857620584856omplex(H,B5) )
              <=> ( groups7754918857620584856omplex(G2,C4) = groups7754918857620584856omplex(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5684_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G2: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B5),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_real(G2,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B5)))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G2,A4) = groups8097168146408367636l_real(H,B5) )
              <=> ( groups8097168146408367636l_real(G2,C4) = groups8097168146408367636l_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5685_sum_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B5: set_int,G2: 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(A4),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B5),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A4)))
               => ( aa_int_real(G2,A3) = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B5)))
                 => ( aa_int_real(H,B3) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G2,A4) = groups8778361861064173332t_real(H,B5) )
              <=> ( groups8778361861064173332t_real(G2,C4) = groups8778361861064173332t_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5686_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G2: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B5),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_real(G2,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B5)))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G2,A4) = groups5808333547571424918x_real(H,B5) )
              <=> ( groups5808333547571424918x_real(G2,C4) = groups5808333547571424918x_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5687_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G2: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B5),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_rat(G2,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B5)))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G2,A4) = groups1300246762558778688al_rat(H,B5) )
              <=> ( groups1300246762558778688al_rat(G2,C4) = groups1300246762558778688al_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5688_sum_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B5: set_int,G2: 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(A4),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B5),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A4)))
               => ( aa_int_rat(G2,A3) = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B5)))
                 => ( aa_int_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G2,A4) = groups3906332499630173760nt_rat(H,B5) )
              <=> ( groups3906332499630173760nt_rat(G2,C4) = groups3906332499630173760nt_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5689_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G2: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B5),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A4)))
               => ( aa_complex_rat(G2,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B5)))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G2,A4) = groups5058264527183730370ex_rat(H,B5) )
              <=> ( groups5058264527183730370ex_rat(G2,C4) = groups5058264527183730370ex_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5690_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G2: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,C4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),C4))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B5),C4))
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),minus_minus_set_real(C4,A4)))
               => ( aa_real_nat(G2,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(C4,B5)))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G2,A4) = groups1935376822645274424al_nat(H,B5) )
              <=> ( groups1935376822645274424al_nat(G2,C4) = groups1935376822645274424al_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5691_sum_Omono__neutral__cong,axiom,
    ! [T4: set_real,S3: set_real,H: fun_real_complex,G2: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(finite_finite_real,S3))
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),minus_minus_set_real(T4,S3)))
             => ( aa_real_complex(H,I3) = zero_zero_complex ) )
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),minus_minus_set_real(S3,T4)))
               => ( aa_real_complex(G2,I3) = zero_zero_complex ) )
           => ( ! [X3: real] :
                  ( pp(aa_set_real_bool(member_real(X3),inf_inf_set_real(S3,T4)))
                 => ( aa_real_complex(G2,X3) = aa_real_complex(H,X3) ) )
             => ( groups5754745047067104278omplex(G2,S3) = groups5754745047067104278omplex(H,T4) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_5692_sum_Omono__neutral__cong,axiom,
    ! [T4: set_int,S3: set_int,H: fun_int_complex,G2: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(finite_finite_int,S3))
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),minus_minus_set_int(T4,S3)))
             => ( aa_int_complex(H,I3) = zero_zero_complex ) )
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),minus_minus_set_int(S3,T4)))
               => ( aa_int_complex(G2,I3) = zero_zero_complex ) )
           => ( ! [X3: int] :
                  ( pp(aa_set_int_bool(member_int(X3),inf_inf_set_int(S3,T4)))
                 => ( aa_int_complex(G2,X3) = aa_int_complex(H,X3) ) )
             => ( groups3049146728041665814omplex(G2,S3) = groups3049146728041665814omplex(H,T4) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_5693_sum_Omono__neutral__cong,axiom,
    ! [T4: set_complex,S3: set_complex,H: fun_complex_complex,G2: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_complex(H,I3) = zero_zero_complex ) )
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),minus_811609699411566653omplex(S3,T4)))
               => ( aa_complex_complex(G2,I3) = zero_zero_complex ) )
           => ( ! [X3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X3),inf_inf_set_complex(S3,T4)))
                 => ( aa_complex_complex(G2,X3) = aa_complex_complex(H,X3) ) )
             => ( groups7754918857620584856omplex(G2,S3) = groups7754918857620584856omplex(H,T4) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_5694_sum_Omono__neutral__cong,axiom,
    ! [T4: set_real,S3: set_real,H: fun_real_real,G2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(finite_finite_real,S3))
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),minus_minus_set_real(T4,S3)))
             => ( aa_real_real(H,I3) = zero_zero_real ) )
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),minus_minus_set_real(S3,T4)))
               => ( aa_real_real(G2,I3) = zero_zero_real ) )
           => ( ! [X3: real] :
                  ( pp(aa_set_real_bool(member_real(X3),inf_inf_set_real(S3,T4)))
                 => ( aa_real_real(G2,X3) = aa_real_real(H,X3) ) )
             => ( groups8097168146408367636l_real(G2,S3) = groups8097168146408367636l_real(H,T4) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_5695_sum_Omono__neutral__cong,axiom,
    ! [T4: set_int,S3: set_int,H: fun_int_real,G2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(finite_finite_int,S3))
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),minus_minus_set_int(T4,S3)))
             => ( aa_int_real(H,I3) = zero_zero_real ) )
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),minus_minus_set_int(S3,T4)))
               => ( aa_int_real(G2,I3) = zero_zero_real ) )
           => ( ! [X3: int] :
                  ( pp(aa_set_int_bool(member_int(X3),inf_inf_set_int(S3,T4)))
                 => ( aa_int_real(G2,X3) = aa_int_real(H,X3) ) )
             => ( groups8778361861064173332t_real(G2,S3) = groups8778361861064173332t_real(H,T4) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_5696_sum_Omono__neutral__cong,axiom,
    ! [T4: set_complex,S3: set_complex,H: fun_complex_real,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_real(H,I3) = zero_zero_real ) )
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),minus_811609699411566653omplex(S3,T4)))
               => ( aa_complex_real(G2,I3) = zero_zero_real ) )
           => ( ! [X3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X3),inf_inf_set_complex(S3,T4)))
                 => ( aa_complex_real(G2,X3) = aa_complex_real(H,X3) ) )
             => ( groups5808333547571424918x_real(G2,S3) = groups5808333547571424918x_real(H,T4) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_5697_sum_Omono__neutral__cong,axiom,
    ! [T4: set_real,S3: set_real,H: fun_real_rat,G2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(finite_finite_real,S3))
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),minus_minus_set_real(T4,S3)))
             => ( aa_real_rat(H,I3) = zero_zero_rat ) )
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),minus_minus_set_real(S3,T4)))
               => ( aa_real_rat(G2,I3) = zero_zero_rat ) )
           => ( ! [X3: real] :
                  ( pp(aa_set_real_bool(member_real(X3),inf_inf_set_real(S3,T4)))
                 => ( aa_real_rat(G2,X3) = aa_real_rat(H,X3) ) )
             => ( groups1300246762558778688al_rat(G2,S3) = groups1300246762558778688al_rat(H,T4) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_5698_sum_Omono__neutral__cong,axiom,
    ! [T4: set_int,S3: set_int,H: fun_int_rat,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(finite_finite_int,S3))
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),minus_minus_set_int(T4,S3)))
             => ( aa_int_rat(H,I3) = zero_zero_rat ) )
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),minus_minus_set_int(S3,T4)))
               => ( aa_int_rat(G2,I3) = zero_zero_rat ) )
           => ( ! [X3: int] :
                  ( pp(aa_set_int_bool(member_int(X3),inf_inf_set_int(S3,T4)))
                 => ( aa_int_rat(G2,X3) = aa_int_rat(H,X3) ) )
             => ( groups3906332499630173760nt_rat(G2,S3) = groups3906332499630173760nt_rat(H,T4) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_5699_sum_Omono__neutral__cong,axiom,
    ! [T4: set_complex,S3: set_complex,H: fun_complex_rat,G2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_rat(H,I3) = zero_zero_rat ) )
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),minus_811609699411566653omplex(S3,T4)))
               => ( aa_complex_rat(G2,I3) = zero_zero_rat ) )
           => ( ! [X3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X3),inf_inf_set_complex(S3,T4)))
                 => ( aa_complex_rat(G2,X3) = aa_complex_rat(H,X3) ) )
             => ( groups5058264527183730370ex_rat(G2,S3) = groups5058264527183730370ex_rat(H,T4) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_5700_sum_Omono__neutral__cong,axiom,
    ! [T4: set_real,S3: set_real,H: fun_real_nat,G2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,T4))
     => ( pp(aa_set_real_bool(finite_finite_real,S3))
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),minus_minus_set_real(T4,S3)))
             => ( aa_real_nat(H,I3) = zero_zero_nat ) )
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),minus_minus_set_real(S3,T4)))
               => ( aa_real_nat(G2,I3) = zero_zero_nat ) )
           => ( ! [X3: real] :
                  ( pp(aa_set_real_bool(member_real(X3),inf_inf_set_real(S3,T4)))
                 => ( aa_real_nat(G2,X3) = aa_real_nat(H,X3) ) )
             => ( groups1935376822645274424al_nat(G2,S3) = groups1935376822645274424al_nat(H,T4) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_5701_unity__coeff__ex,axiom,
    ! [P2: fun_complex_bool,L: complex] :
      ( ? [X4: complex] : pp(aa_complex_bool(P2,aa_complex_complex(times_times_complex(L),X4)))
    <=> ? [X4: complex] :
          ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,L),aa_complex_complex(plus_plus_complex(X4),zero_zero_complex)))
          & pp(aa_complex_bool(P2,X4)) ) ) ).

% unity_coeff_ex
tff(fact_5702_unity__coeff__ex,axiom,
    ! [P2: fun_real_bool,L: real] :
      ( ? [X4: real] : pp(aa_real_bool(P2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X4)))
    <=> ? [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),aa_real_real(plus_plus_real(X4),zero_zero_real)))
          & pp(aa_real_bool(P2,X4)) ) ) ).

% unity_coeff_ex
tff(fact_5703_unity__coeff__ex,axiom,
    ! [P2: fun_rat_bool,L: rat] :
      ( ? [X4: rat] : pp(aa_rat_bool(P2,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,L),X4)))
    <=> ? [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,L),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),zero_zero_rat)))
          & pp(aa_rat_bool(P2,X4)) ) ) ).

% unity_coeff_ex
tff(fact_5704_unity__coeff__ex,axiom,
    ! [P2: fun_nat_bool,L: nat] :
      ( ? [X4: nat] : pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X4)))
    <=> ? [X4: 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,X4),zero_zero_nat)))
          & pp(aa_nat_bool(P2,X4)) ) ) ).

% unity_coeff_ex
tff(fact_5705_unity__coeff__ex,axiom,
    ! [P2: fun_int_bool,L: int] :
      ( ? [X4: int] : pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(times_times_int,L),X4)))
    <=> ? [X4: 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,X4),zero_zero_int)))
          & pp(aa_int_bool(P2,X4)) ) ) ).

% unity_coeff_ex
tff(fact_5706_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) ) ) ).

% unit_dvdE
tff(fact_5707_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 )
         => ! [C: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) ) ) ).

% unit_dvdE
tff(fact_5708_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 )
         => ! [C: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C) ) ) ).

% unit_dvdE
tff(fact_5709_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C2: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C2 != 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,C2),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),C2) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_5710_dvd__div__div__eq__mult,axiom,
    ! [A: int,C2: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C2 != 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,C2),D))
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D),C2) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_5711_dvd__div__iff__mult,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( C2 != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),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),C2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_5712_dvd__div__iff__mult,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( C2 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_5713_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C2: 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)),C2))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_5714_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C2: 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(aa_int_fun_int_int(divide_divide_int,A),B)),C2))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_5715_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C2: 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) = C2 )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_5716_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = C2 )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_5717_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_5718_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_5719_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(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_5720_inf__period_I4_J,axiom,
    ! [D: real,D3: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D3))
     => ! [X5: real,K4: real] :
          ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(plus_plus_real(X5),T)))
        <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(X5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3))),T))) ) ) ).

% inf_period(4)
tff(fact_5721_inf__period_I4_J,axiom,
    ! [D: rat,D3: rat,T: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),D3))
     => ! [X5: rat,K4: rat] :
          ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),T)))
        <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3))),T))) ) ) ).

% inf_period(4)
tff(fact_5722_inf__period_I4_J,axiom,
    ! [D: int,D3: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
     => ! [X5: 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,X5),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,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))),T))) ) ) ).

% inf_period(4)
tff(fact_5723_inf__period_I3_J,axiom,
    ! [D: real,D3: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D3))
     => ! [X5: real,K4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(plus_plus_real(X5),T)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(X5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3))),T))) ) ) ).

% inf_period(3)
tff(fact_5724_inf__period_I3_J,axiom,
    ! [D: rat,D3: rat,T: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),D3))
     => ! [X5: rat,K4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),T)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3))),T))) ) ) ).

% inf_period(3)
tff(fact_5725_inf__period_I3_J,axiom,
    ! [D: int,D3: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
     => ! [X5: 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,X5),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,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))),T))) ) ) ).

% inf_period(3)
tff(fact_5726_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( ( divide6298287555418463151nteger(A,B) = C2 )
      <=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B) ) ) ) ).

% unit_eq_div1
tff(fact_5727_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C2: 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) = C2 )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B) ) ) ) ).

% unit_eq_div1
tff(fact_5728_unit__eq__div1,axiom,
    ! [B: int,A: int,C2: 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(divide_divide_int,A),B) = C2 )
      <=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B) ) ) ) ).

% unit_eq_div1
tff(fact_5729_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( ( A = divide6298287555418463151nteger(C2,B) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = C2 ) ) ) ).

% unit_eq_div2
tff(fact_5730_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C2: 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,C2),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C2 ) ) ) ).

% unit_eq_div2
tff(fact_5731_unit__eq__div2,axiom,
    ! [B: int,A: int,C2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( ( A = aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),B) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = C2 ) ) ) ).

% unit_eq_div2
tff(fact_5732_div__mult__unit2,axiom,
    ! [C2: code_integer,B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C2),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C2) ) ) ) ).

% div_mult_unit2
tff(fact_5733_div__mult__unit2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2) ) ) ) ).

% div_mult_unit2
tff(fact_5734_div__mult__unit2,axiom,
    ! [C2: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2) ) ) ) ).

% div_mult_unit2
tff(fact_5735_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C2) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2),B) ) ) ).

% unit_div_commute
tff(fact_5736_unit__div__commute,axiom,
    ! [B: nat,A: nat,C2: 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)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),B) ) ) ).

% unit_div_commute
tff(fact_5737_unit__div__commute,axiom,
    ! [B: int,A: int,C2: 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(aa_int_fun_int_int(divide_divide_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),B) ) ) ).

% unit_div_commute
tff(fact_5738_unit__div__mult__swap,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C2),one_one_Code_integer))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C2)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C2) ) ) ).

% unit_div_mult_swap
tff(fact_5739_unit__div__mult__swap,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C2),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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) ) ) ).

% unit_div_mult_swap
tff(fact_5740_unit__div__mult__swap,axiom,
    ! [C2: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C2),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2) ) ) ).

% unit_div_mult_swap
tff(fact_5741_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C2: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C2),one_one_Code_integer))
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C2) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_5742_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C2: 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,C2),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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_5743_is__unit__div__mult2__eq,axiom,
    ! [B: int,C2: 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,C2),one_one_int))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_5744_is__unit__power__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),one_one_Code_integer))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5745_is__unit__power__iff,axiom,
    ! [A: nat,N: 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),N)),one_one_nat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5746_is__unit__power__iff,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),one_one_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5747_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_5748_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_5749_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(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_5750_unit__imp__mod__eq__0,axiom,
    ! [B: code_natural,A: code_natural] :
      ( dvd_dvd_Code_natural(B,one_one_Code_natural)
     => ( modulo8411746178871703098atural(A,B) = zero_z2226904508553997617atural ) ) ).

% unit_imp_mod_eq_0
tff(fact_5751_pochhammer__nonneg,axiom,
    ! [X: real,N: 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(aa_real_fun_nat_real(comm_s7457072308508201937r_real,X),N))) ) ).

% pochhammer_nonneg
tff(fact_5752_pochhammer__nonneg,axiom,
    ! [X: rat,N: 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),N))) ) ).

% pochhammer_nonneg
tff(fact_5753_pochhammer__nonneg,axiom,
    ! [X: nat,N: 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),N))) ) ).

% pochhammer_nonneg
tff(fact_5754_pochhammer__nonneg,axiom,
    ! [X: int,N: 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),N))) ) ).

% pochhammer_nonneg
tff(fact_5755_dvd__imp__le,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K2),N))
     => ( 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(ord_less_eq_nat,K2),N)) ) ) ).

% dvd_imp_le
tff(fact_5756_dvd__mult__cancel,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N)) ) ) ).

% dvd_mult_cancel
tff(fact_5757_nat__mult__dvd__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_5758_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(zero_z3403309356797280102nteger),N) = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(zero_z3403309356797280102nteger),N) = zero_z3403309356797280102nteger ) ) ) ).

% pochhammer_0_left
tff(fact_5759_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% pochhammer_0_left
tff(fact_5760_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),N) = zero_zero_real ) ) ) ).

% pochhammer_0_left
tff(fact_5761_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).

% pochhammer_0_left
tff(fact_5762_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).

% pochhammer_0_left
tff(fact_5763_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = zero_zero_int ) ) ) ).

% pochhammer_0_left
tff(fact_5764_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D5: nat,X3: 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),X3) = 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_5765_mod__greater__zero__iff__not__dvd,axiom,
    ! [M2: nat,N: 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,M2),N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M2)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_5766_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M2: nat,Q2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_5767_sum__power__add,axiom,
    ! [X: complex,M2: nat,I5: set_nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_gq(X),M2)),I5) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M2)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),I5)) ).

% sum_power_add
tff(fact_5768_sum__power__add,axiom,
    ! [X: rat,M2: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_gr(X),M2),I5) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),M2)),groups2906978787729119204at_rat(power_power_rat(X),I5)) ).

% sum_power_add
tff(fact_5769_sum__power__add,axiom,
    ! [X: int,M2: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_gs(X),M2),I5) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),M2)),groups3539618377306564664at_int(power_power_int(X),I5)) ).

% sum_power_add
tff(fact_5770_sum__power__add,axiom,
    ! [X: real,M2: nat,I5: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_gt(X),M2)),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M2)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),I5)) ).

% sum_power_add
tff(fact_5771_sum__mono2,axiom,
    ! [B5: set_real,A4: set_real,F2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B5))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B5,A4)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F2,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F2,A4)),groups8097168146408367636l_real(F2,B5))) ) ) ) ).

% sum_mono2
tff(fact_5772_sum__mono2,axiom,
    ! [B5: set_int,A4: set_int,F2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,B5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B5))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B5,A4)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F2,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F2,A4)),groups8778361861064173332t_real(F2,B5))) ) ) ) ).

% sum_mono2
tff(fact_5773_sum__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B5))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B5,A4)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F2,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F2,A4)),groups5808333547571424918x_real(F2,B5))) ) ) ) ).

% sum_mono2
tff(fact_5774_sum__mono2,axiom,
    ! [B5: set_real,A4: set_real,F2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B5))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B5,A4)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F2,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F2,A4)),groups1300246762558778688al_rat(F2,B5))) ) ) ) ).

% sum_mono2
tff(fact_5775_sum__mono2,axiom,
    ! [B5: set_int,A4: set_int,F2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,B5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B5))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B5,A4)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F2,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F2,A4)),groups3906332499630173760nt_rat(F2,B5))) ) ) ) ).

% sum_mono2
tff(fact_5776_sum__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B5))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B5,A4)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F2,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F2,A4)),groups5058264527183730370ex_rat(F2,B5))) ) ) ) ).

% sum_mono2
tff(fact_5777_sum__mono2,axiom,
    ! [B5: set_real,A4: set_real,F2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B5))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B5,A4)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F2,B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F2,A4)),groups1935376822645274424al_nat(F2,B5))) ) ) ) ).

% sum_mono2
tff(fact_5778_sum__mono2,axiom,
    ! [B5: set_int,A4: set_int,F2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B5))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B5,A4)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F2,B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F2,A4)),groups4541462559716669496nt_nat(F2,B5))) ) ) ) ).

% sum_mono2
tff(fact_5779_sum__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B5))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B5,A4)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F2,B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups5693394587270226106ex_nat(F2,A4)),groups5693394587270226106ex_nat(F2,B5))) ) ) ) ).

% sum_mono2
tff(fact_5780_sum__mono2,axiom,
    ! [B5: set_real,A4: set_real,F2: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B5))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B5,A4)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F2,B3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F2,A4)),groups1932886352136224148al_int(F2,B5))) ) ) ) ).

% sum_mono2
tff(fact_5781_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),inf_inf_set_int(A4,B5)))
             => ( aa_int_complex(G2,X3) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G2,sup_sup_set_int(A4,B5)) = aa_complex_complex(plus_plus_complex(groups3049146728041665814omplex(G2,A4)),groups3049146728041665814omplex(G2,B5)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_5782_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),inf_inf_set_complex(A4,B5)))
             => ( aa_complex_complex(G2,X3) = zero_zero_complex ) )
         => ( groups7754918857620584856omplex(G2,sup_sup_set_complex(A4,B5)) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G2,A4)),groups7754918857620584856omplex(G2,B5)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_5783_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),inf_inf_set_int(A4,B5)))
             => ( aa_int_real(G2,X3) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G2,sup_sup_set_int(A4,B5)) = aa_real_real(plus_plus_real(groups8778361861064173332t_real(G2,A4)),groups8778361861064173332t_real(G2,B5)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_5784_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),inf_inf_set_complex(A4,B5)))
             => ( aa_complex_real(G2,X3) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G2,sup_sup_set_complex(A4,B5)) = aa_real_real(plus_plus_real(groups5808333547571424918x_real(G2,A4)),groups5808333547571424918x_real(G2,B5)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_5785_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),inf_inf_set_int(A4,B5)))
             => ( aa_int_rat(G2,X3) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G2,sup_sup_set_int(A4,B5)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(G2,A4)),groups3906332499630173760nt_rat(G2,B5)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_5786_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),inf_inf_set_complex(A4,B5)))
             => ( aa_complex_rat(G2,X3) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G2,sup_sup_set_complex(A4,B5)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G2,A4)),groups5058264527183730370ex_rat(G2,B5)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_5787_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),inf_inf_set_int(A4,B5)))
             => ( aa_int_nat(G2,X3) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G2,sup_sup_set_int(A4,B5)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G2,A4)),groups4541462559716669496nt_nat(G2,B5)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_5788_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),inf_inf_set_complex(A4,B5)))
             => ( aa_complex_nat(G2,X3) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G2,sup_sup_set_complex(A4,B5)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G2,A4)),groups5693394587270226106ex_nat(G2,B5)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_5789_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),inf_inf_set_complex(A4,B5)))
             => ( aa_complex_int(G2,X3) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G2,sup_sup_set_complex(A4,B5)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G2,A4)),groups5690904116761175830ex_int(G2,B5)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_5790_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B5: set_nat,G2: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5)))
             => ( aa_nat_complex(G2,X3) = zero_zero_complex ) )
         => ( aa_set_nat_complex(groups2073611262835488442omplex(G2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5)) = aa_complex_complex(plus_plus_complex(aa_set_nat_complex(groups2073611262835488442omplex(G2),A4)),aa_set_nat_complex(groups2073611262835488442omplex(G2),B5)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_5791_sum_OatLeastAtMost__rev,axiom,
    ! [G2: fun_nat_nat,N: nat,M2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_gu(G2),N),M2)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M2)) ).

% sum.atLeastAtMost_rev
tff(fact_5792_sum_OatLeastAtMost__rev,axiom,
    ! [G2: fun_nat_real,N: nat,M2: nat] : aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_gv(G2),N),M2)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M2)) ).

% sum.atLeastAtMost_rev
tff(fact_5793_finite__divisors__nat,axiom,
    ! [M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_fr,M2)))) ) ).

% finite_divisors_nat
tff(fact_5794_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aTP_Lamm_ei(A)),aa_nat_complex(A,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_5795_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : pp(aa_real_bool(sums_real(aTP_Lamm_ej(A)),aa_nat_real(A,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_5796_sum__shift__lb__Suc0__0,axiom,
    ! [F2: fun_nat_complex,K2: nat] :
      ( ( aa_nat_complex(F2,zero_zero_nat) = zero_zero_complex )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K2)) = aa_set_nat_complex(groups2073611262835488442omplex(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5797_sum__shift__lb__Suc0__0,axiom,
    ! [F2: fun_nat_rat,K2: nat] :
      ( ( aa_nat_rat(F2,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K2)) = groups2906978787729119204at_rat(F2,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5798_sum__shift__lb__Suc0__0,axiom,
    ! [F2: fun_nat_int,K2: nat] :
      ( ( aa_nat_int(F2,zero_zero_nat) = zero_zero_int )
     => ( groups3539618377306564664at_int(F2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K2)) = groups3539618377306564664at_int(F2,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5799_sum__shift__lb__Suc0__0,axiom,
    ! [F2: fun_nat_nat,K2: nat] :
      ( ( aa_nat_nat(F2,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K2)) = aa_set_nat_nat(groups3542108847815614940at_nat(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5800_sum__shift__lb__Suc0__0,axiom,
    ! [F2: fun_nat_real,K2: nat] :
      ( ( aa_nat_real(F2,zero_zero_nat) = zero_zero_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K2)) = aa_set_nat_real(groups6591440286371151544t_real(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5801_sum_OatLeast0__atMost__Suc,axiom,
    ! [G2: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_rat(G2,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5802_sum_OatLeast0__atMost__Suc,axiom,
    ! [G2: fun_nat_int,N: nat] : groups3539618377306564664at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_int(G2,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5803_sum_OatLeast0__atMost__Suc,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_nat(G2,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5804_sum_OatLeast0__atMost__Suc,axiom,
    ! [G2: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_real(G2,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5805_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G2,M2)),groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5806_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( groups3539618377306564664at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G2,M2)),groups3539618377306564664at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5807_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G2,M2)),aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5808_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_real_real(plus_plus_real(aa_nat_real(G2,M2)),aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5809_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G2,aa_nat_nat(suc,N))),groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5810_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( groups3539618377306564664at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G2,aa_nat_nat(suc,N))),groups3539618377306564664at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5811_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G2,aa_nat_nat(suc,N))),aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5812_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_real_real(plus_plus_real(aa_nat_real(G2,aa_nat_nat(suc,N))),aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5813_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_5814_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_5815_is__unitE,axiom,
    ! [A: code_integer,C2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B3: code_integer] :
              ( ( B3 != zero_z3403309356797280102nteger )
             => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B3),one_one_Code_integer))
               => ( ( divide6298287555418463151nteger(one_one_Code_integer,A) = B3 )
                 => ( ( divide6298287555418463151nteger(one_one_Code_integer,B3) = A )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B3) = one_one_Code_integer )
                     => ( divide6298287555418463151nteger(C2,A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_5816_is__unitE,axiom,
    ! [A: nat,C2: 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,C2),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_5817_is__unitE,axiom,
    ! [A: int,C2: 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(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B3 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B3) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B3) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C2),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_5818_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_5819_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_5820_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(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_5821_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_5822_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_5823_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(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_5824_evenE,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(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_5825_evenE,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(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_5826_dvd__power__iff,axiom,
    ! [X: code_integer,M2: nat,N: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),M2)),aa_nat_Code_integer(power_8256067586552552935nteger(X),N)))
      <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),one_one_Code_integer))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ) ).

% dvd_power_iff
tff(fact_5827_dvd__power__iff,axiom,
    ! [X: nat,M2: nat,N: 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),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)))
      <=> ( 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,M2),N)) ) ) ) ).

% dvd_power_iff
tff(fact_5828_dvd__power__iff,axiom,
    ! [X: int,M2: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X),M2)),aa_nat_int(power_power_int(X),N)))
      <=> ( 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,M2),N)) ) ) ) ).

% dvd_power_iff
tff(fact_5829_dvd__power,axiom,
    ! [N: nat,X: code_integer] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_Code_integer ) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),aa_nat_Code_integer(power_8256067586552552935nteger(X),N))) ) ).

% dvd_power
tff(fact_5830_dvd__power,axiom,
    ! [N: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( 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),N))) ) ).

% dvd_power
tff(fact_5831_dvd__power,axiom,
    ! [N: nat,X: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_real ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ).

% dvd_power
tff(fact_5832_dvd__power,axiom,
    ! [N: nat,X: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_int ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_nat_int(power_power_int(X),N))) ) ).

% dvd_power
tff(fact_5833_dvd__power,axiom,
    ! [N: nat,X: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_complex ) )
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),aa_nat_complex(power_power_complex(X),N))) ) ).

% dvd_power
tff(fact_5834_sum__strict__mono2,axiom,
    ! [B5: set_real,A4: set_real,B: real,F2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B5))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B5,A4)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F2,B)))
           => ( ! [X3: real] :
                  ( pp(aa_set_real_bool(member_real(X3),B5))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F2,X3))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F2,A4)),groups8097168146408367636l_real(F2,B5))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5835_sum__strict__mono2,axiom,
    ! [B5: set_int,A4: set_int,B: int,F2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,B5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B5))
       => ( pp(aa_set_int_bool(member_int(B),minus_minus_set_int(B5,A4)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F2,B)))
           => ( ! [X3: int] :
                  ( pp(aa_set_int_bool(member_int(X3),B5))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F2,X3))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F2,A4)),groups8778361861064173332t_real(F2,B5))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5836_sum__strict__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,B: complex,F2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B5))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B5,A4)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F2,B)))
           => ( ! [X3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X3),B5))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F2,X3))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F2,A4)),groups5808333547571424918x_real(F2,B5))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5837_sum__strict__mono2,axiom,
    ! [B5: set_real,A4: set_real,B: real,F2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B5))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B5,A4)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F2,B)))
           => ( ! [X3: real] :
                  ( pp(aa_set_real_bool(member_real(X3),B5))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F2,X3))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F2,A4)),groups1300246762558778688al_rat(F2,B5))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5838_sum__strict__mono2,axiom,
    ! [B5: set_int,A4: set_int,B: int,F2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,B5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B5))
       => ( pp(aa_set_int_bool(member_int(B),minus_minus_set_int(B5,A4)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F2,B)))
           => ( ! [X3: int] :
                  ( pp(aa_set_int_bool(member_int(X3),B5))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F2,X3))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F2,A4)),groups3906332499630173760nt_rat(F2,B5))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5839_sum__strict__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,B: complex,F2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B5))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B5,A4)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F2,B)))
           => ( ! [X3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X3),B5))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F2,X3))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F2,A4)),groups5058264527183730370ex_rat(F2,B5))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5840_sum__strict__mono2,axiom,
    ! [B5: set_real,A4: set_real,B: real,F2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B5))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B5,A4)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F2,B)))
           => ( ! [X3: real] :
                  ( pp(aa_set_real_bool(member_real(X3),B5))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F2,X3))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups1935376822645274424al_nat(F2,A4)),groups1935376822645274424al_nat(F2,B5))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5841_sum__strict__mono2,axiom,
    ! [B5: set_int,A4: set_int,B: int,F2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B5))
       => ( pp(aa_set_int_bool(member_int(B),minus_minus_set_int(B5,A4)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F2,B)))
           => ( ! [X3: int] :
                  ( pp(aa_set_int_bool(member_int(X3),B5))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F2,X3))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F2,A4)),groups4541462559716669496nt_nat(F2,B5))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5842_sum__strict__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,B: complex,F2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B5))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B5,A4)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F2,B)))
           => ( ! [X3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X3),B5))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F2,X3))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F2,A4)),groups5693394587270226106ex_nat(F2,B5))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5843_sum__strict__mono2,axiom,
    ! [B5: set_real,A4: set_real,B: real,F2: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B5))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B5,A4)))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(F2,B)))
           => ( ! [X3: real] :
                  ( pp(aa_set_real_bool(member_real(X3),B5))
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F2,X3))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups1932886352136224148al_int(F2,A4)),groups1932886352136224148al_int(F2,B5))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5844_member__le__sum,axiom,
    ! [I2: complex,A4: set_complex,F2: fun_complex_real] :
      ( pp(aa_set_complex_bool(member_complex(I2),A4))
     => ( ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(A4,insert_complex(I2,bot_bot_set_complex))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F2,X3))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F2,I2)),groups5808333547571424918x_real(F2,A4))) ) ) ) ).

% member_le_sum
tff(fact_5845_member__le__sum,axiom,
    ! [I2: int,A4: set_int,F2: fun_int_real] :
      ( pp(aa_set_int_bool(member_int(I2),A4))
     => ( ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(A4,insert_int(I2,bot_bot_set_int))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F2,X3))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F2,I2)),groups8778361861064173332t_real(F2,A4))) ) ) ) ).

% member_le_sum
tff(fact_5846_member__le__sum,axiom,
    ! [I2: real,A4: set_real,F2: fun_real_real] :
      ( pp(aa_set_real_bool(member_real(I2),A4))
     => ( ! [X3: real] :
            ( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(A4,insert_real(I2,bot_bot_set_real))))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F2,X3))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F2,I2)),groups8097168146408367636l_real(F2,A4))) ) ) ) ).

% member_le_sum
tff(fact_5847_member__le__sum,axiom,
    ! [I2: complex,A4: set_complex,F2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(member_complex(I2),A4))
     => ( ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(A4,insert_complex(I2,bot_bot_set_complex))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F2,X3))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F2,I2)),groups5058264527183730370ex_rat(F2,A4))) ) ) ) ).

% member_le_sum
tff(fact_5848_member__le__sum,axiom,
    ! [I2: int,A4: set_int,F2: fun_int_rat] :
      ( pp(aa_set_int_bool(member_int(I2),A4))
     => ( ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(A4,insert_int(I2,bot_bot_set_int))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F2,X3))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F2,I2)),groups3906332499630173760nt_rat(F2,A4))) ) ) ) ).

% member_le_sum
tff(fact_5849_member__le__sum,axiom,
    ! [I2: real,A4: set_real,F2: fun_real_rat] :
      ( pp(aa_set_real_bool(member_real(I2),A4))
     => ( ! [X3: real] :
            ( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(A4,insert_real(I2,bot_bot_set_real))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F2,X3))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F2,I2)),groups1300246762558778688al_rat(F2,A4))) ) ) ) ).

% member_le_sum
tff(fact_5850_member__le__sum,axiom,
    ! [I2: nat,A4: set_nat,F2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(member_nat(I2),A4))
     => ( ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(I2,bot_bot_set_nat))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F2,X3))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,I2)),groups2906978787729119204at_rat(F2,A4))) ) ) ) ).

% member_le_sum
tff(fact_5851_member__le__sum,axiom,
    ! [I2: complex,A4: set_complex,F2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(member_complex(I2),A4))
     => ( ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(A4,insert_complex(I2,bot_bot_set_complex))))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F2,X3))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F2,I2)),groups5693394587270226106ex_nat(F2,A4))) ) ) ) ).

% member_le_sum
tff(fact_5852_member__le__sum,axiom,
    ! [I2: int,A4: set_int,F2: fun_int_nat] :
      ( pp(aa_set_int_bool(member_int(I2),A4))
     => ( ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(A4,insert_int(I2,bot_bot_set_int))))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F2,X3))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F2,I2)),groups4541462559716669496nt_nat(F2,A4))) ) ) ) ).

% member_le_sum
tff(fact_5853_member__le__sum,axiom,
    ! [I2: real,A4: set_real,F2: fun_real_nat] :
      ( pp(aa_set_real_bool(member_real(I2),A4))
     => ( ! [X3: real] :
            ( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(A4,insert_real(I2,bot_bot_set_real))))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F2,X3))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F2,I2)),groups1935376822645274424al_nat(F2,A4))) ) ) ) ).

% member_le_sum
tff(fact_5854_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_5855_sum__bounded__above__strict,axiom,
    ! [A4: set_real,F2: fun_real_rat,K5: rat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F2,I3)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_real(A4)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F2,A4)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_real(A4))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_5856_sum__bounded__above__strict,axiom,
    ! [A4: set_nat,F2: fun_nat_rat,K5: rat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F2,I3)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A4)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F2,A4)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A4))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_5857_sum__bounded__above__strict,axiom,
    ! [A4: set_complex,F2: fun_complex_rat,K5: rat] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F2,I3)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_complex(A4)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F2,A4)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_complex(A4))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_5858_sum__bounded__above__strict,axiom,
    ! [A4: set_int,F2: fun_int_rat,K5: rat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F2,I3)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_int(A4)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F2,A4)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_int(A4))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_5859_sum__bounded__above__strict,axiom,
    ! [A4: set_real,F2: fun_real_real,K5: real] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F2,I3)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_real(A4)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F2,A4)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_real(A4))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_5860_sum__bounded__above__strict,axiom,
    ! [A4: set_complex,F2: fun_complex_real,K5: real] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F2,I3)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_complex(A4)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F2,A4)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_complex(A4))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_5861_sum__bounded__above__strict,axiom,
    ! [A4: set_int,F2: fun_int_real,K5: real] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F2,I3)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_int(A4)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F2,A4)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_int(A4))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_5862_sum__bounded__above__strict,axiom,
    ! [A4: set_real,F2: fun_real_int,K5: int] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F2,I3)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_real(A4)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups1932886352136224148al_int(F2,A4)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_real(A4))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_5863_sum__bounded__above__strict,axiom,
    ! [A4: set_nat,F2: fun_nat_int,K5: int] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F2,I3)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A4)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F2,A4)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(finite_card_nat,A4))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_5864_sum__bounded__above__strict,axiom,
    ! [A4: set_complex,F2: fun_complex_int,K5: int] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_complex_int(F2,I3)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),finite_card_complex(A4)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups5690904116761175830ex_int(F2,A4)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_complex(A4))),K5))) ) ) ).

% sum_bounded_above_strict
tff(fact_5865_sum__bounded__above__divide,axiom,
    ! [A4: set_complex,F2: fun_complex_real,K5: real] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F2,I3)),aa_real_real(divide_divide_real(K5),aa_nat_real(semiri5074537144036343181t_real,finite_card_complex(A4))))) )
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
       => ( ( A4 != bot_bot_set_complex )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F2,A4)),K5)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_5866_sum__bounded__above__divide,axiom,
    ! [A4: set_int,F2: fun_int_real,K5: real] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F2,I3)),aa_real_real(divide_divide_real(K5),aa_nat_real(semiri5074537144036343181t_real,finite_card_int(A4))))) )
     => ( pp(aa_set_int_bool(finite_finite_int,A4))
       => ( ( A4 != bot_bot_set_int )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F2,A4)),K5)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_5867_sum__bounded__above__divide,axiom,
    ! [A4: set_real,F2: fun_real_real,K5: real] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F2,I3)),aa_real_real(divide_divide_real(K5),aa_nat_real(semiri5074537144036343181t_real,finite_card_real(A4))))) )
     => ( pp(aa_set_real_bool(finite_finite_real,A4))
       => ( ( A4 != bot_bot_set_real )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F2,A4)),K5)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_5868_sum__bounded__above__divide,axiom,
    ! [A4: set_complex,F2: fun_complex_rat,K5: rat] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F2,I3)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,K5),aa_nat_rat(semiri681578069525770553at_rat,finite_card_complex(A4))))) )
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
       => ( ( A4 != bot_bot_set_complex )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F2,A4)),K5)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_5869_sum__bounded__above__divide,axiom,
    ! [A4: set_nat,F2: fun_nat_rat,K5: rat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,I3)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,K5),aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A4))))) )
     => ( pp(aa_set_nat_bool(finite_finite_nat,A4))
       => ( ( A4 != bot_bot_set_nat )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F2,A4)),K5)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_5870_sum__bounded__above__divide,axiom,
    ! [A4: set_int,F2: fun_int_rat,K5: rat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F2,I3)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,K5),aa_nat_rat(semiri681578069525770553at_rat,finite_card_int(A4))))) )
     => ( pp(aa_set_int_bool(finite_finite_int,A4))
       => ( ( A4 != bot_bot_set_int )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F2,A4)),K5)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_5871_sum__bounded__above__divide,axiom,
    ! [A4: set_real,F2: fun_real_rat,K5: rat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F2,I3)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,K5),aa_nat_rat(semiri681578069525770553at_rat,finite_card_real(A4))))) )
     => ( pp(aa_set_real_bool(finite_finite_real,A4))
       => ( ( A4 != bot_bot_set_real )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F2,A4)),K5)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_5872_sum__bounded__above__divide,axiom,
    ! [A4: set_nat,F2: fun_nat_real,K5: real] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F2,I3)),aa_real_real(divide_divide_real(K5),aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(finite_card_nat,A4))))) )
     => ( pp(aa_set_nat_bool(finite_finite_nat,A4))
       => ( ( A4 != bot_bot_set_nat )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F2),A4)),K5)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_5873_sum__bounded__above__divide,axiom,
    ! [A4: set_list_nat,F2: fun_list_nat_real,K5: real] :
      ( ! [I3: list_nat] :
          ( member_list_nat(I3,A4)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_list_nat_real(F2,I3)),aa_real_real(divide_divide_real(K5),aa_nat_real(semiri5074537144036343181t_real,finite_card_list_nat(A4))))) )
     => ( finite8100373058378681591st_nat(A4)
       => ( ( A4 != bot_bot_set_list_nat )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8399112307953289288t_real(F2,A4)),K5)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_5874_sum__bounded__above__divide,axiom,
    ! [A4: set_set_nat,F2: fun_set_nat_real,K5: real] :
      ( ! [I3: set_nat] :
          ( member_set_nat(I3,A4)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(F2,I3)),aa_real_real(divide_divide_real(K5),aa_nat_real(semiri5074537144036343181t_real,finite_card_set_nat(A4))))) )
     => ( finite1152437895449049373et_nat(A4)
       => ( ( A4 != bot_bot_set_set_nat )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5107569545109728110t_real(F2,A4)),K5)) ) ) ) ).

% sum_bounded_above_divide
tff(fact_5875_pochhammer__rec,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer)),N)) ).

% pochhammer_rec
tff(fact_5876_pochhammer__rec,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),N)) ).

% pochhammer_rec
tff(fact_5877_pochhammer__rec,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(A),one_one_real)),N)) ).

% pochhammer_rec
tff(fact_5878_pochhammer__rec,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),N)) ).

% pochhammer_rec
tff(fact_5879_pochhammer__rec,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = 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)),N)) ).

% pochhammer_rec
tff(fact_5880_pochhammer__rec,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = 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)),N)) ).

% pochhammer_rec
tff(fact_5881_dvd__mult__cancel1,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)),M2))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_5882_dvd__mult__cancel2,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M2)),M2))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_5883_pochhammer__rec_H,axiom,
    ! [Z2: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z2),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z2),aa_nat_rat(semiri681578069525770553at_rat,N))),aa_nat_rat(comm_s4028243227959126397er_rat(Z2),N)) ).

% pochhammer_rec'
tff(fact_5884_pochhammer__rec_H,axiom,
    ! [Z2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z2),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(plus_plus_real(Z2),aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z2),N)) ).

% pochhammer_rec'
tff(fact_5885_pochhammer__rec_H,axiom,
    ! [Z2: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z2),aa_nat_nat(suc,N)) = 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,N))),aa_nat_int(comm_s4660882817536571857er_int(Z2),N)) ).

% pochhammer_rec'
tff(fact_5886_pochhammer__rec_H,axiom,
    ! [Z2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),aa_nat_nat(suc,N)) = 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,N))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),N)) ).

% pochhammer_rec'
tff(fact_5887_pochhammer__Suc,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(A),N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% pochhammer_Suc
tff(fact_5888_pochhammer__Suc,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N)),aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% pochhammer_Suc
tff(fact_5889_pochhammer__Suc,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% pochhammer_Suc
tff(fact_5890_pochhammer__Suc,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).

% pochhammer_Suc
tff(fact_5891_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),M2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),Q2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),M2)),Q2)))) ) ) ) ).

% dvd_minus_add
tff(fact_5892_power__dvd__imp__le,axiom,
    ! [I2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),I2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% power_dvd_imp_le
tff(fact_5893_pochhammer__product_H,axiom,
    ! [Z2: rat,N: nat,M2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z2),N)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z2),aa_nat_rat(semiri681578069525770553at_rat,N))),M2)) ).

% pochhammer_product'
tff(fact_5894_pochhammer__product_H,axiom,
    ! [Z2: real,N: nat,M2: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z2),N)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(Z2),aa_nat_real(semiri5074537144036343181t_real,N))),M2)) ).

% pochhammer_product'
tff(fact_5895_pochhammer__product_H,axiom,
    ! [Z2: int,N: nat,M2: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z2),N)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),aa_nat_int(semiri1314217659103216013at_int,N))),M2)) ).

% pochhammer_product'
tff(fact_5896_pochhammer__product_H,axiom,
    ! [Z2: nat,N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),N)),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,N))),M2)) ).

% pochhammer_product'
tff(fact_5897_mod__nat__eqI,axiom,
    ! [R: nat,N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,R),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R),M2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),R)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N) = R ) ) ) ) ).

% mod_nat_eqI
tff(fact_5898_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_rat(G2,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G2,M2)),groups2906978787729119204at_rat(aTP_Lamm_gw(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5899_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_int(G2,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G2,M2)),groups3539618377306564664at_int(aTP_Lamm_gx(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5900_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_nat(G2,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G2,M2)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gm(G2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5901_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_real(G2,aa_nat_nat(suc,N))) = aa_real_real(plus_plus_real(aa_nat_real(G2,M2)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gn(G2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5902_sum__Suc__diff,axiom,
    ! [M2: nat,N: nat,F2: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( groups2906978787729119204at_rat(aTP_Lamm_gy(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F2,aa_nat_nat(suc,N))),aa_nat_rat(F2,M2)) ) ) ).

% sum_Suc_diff
tff(fact_5903_sum__Suc__diff,axiom,
    ! [M2: nat,N: nat,F2: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( groups3539618377306564664at_int(aTP_Lamm_gz(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F2,aa_nat_nat(suc,N))),aa_nat_int(F2,M2)) ) ) ).

% sum_Suc_diff
tff(fact_5904_sum__Suc__diff,axiom,
    ! [M2: nat,N: nat,F2: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ha(F2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_real_real(minus_minus_real(aa_nat_real(F2,aa_nat_nat(suc,N))),aa_nat_real(F2,M2)) ) ) ).

% sum_Suc_diff
tff(fact_5905_sum__norm__bound,axiom,
    ! [S3: set_real,F2: fun_real_complex,K5: real] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),S3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(F2,X3))),K5)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups5754745047067104278omplex(F2,S3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_real(S3))),K5))) ) ).

% sum_norm_bound
tff(fact_5906_sum__norm__bound,axiom,
    ! [S3: set_nat,F2: fun_nat_complex,K5: real] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),S3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F2,X3))),K5)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F2),S3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(finite_card_nat,S3))),K5))) ) ).

% sum_norm_bound
tff(fact_5907_sum__norm__bound,axiom,
    ! [S3: set_complex,F2: fun_complex_complex,K5: real] :
      ( ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),S3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(F2,X3))),K5)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F2,S3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_complex(S3))),K5))) ) ).

% sum_norm_bound
tff(fact_5908_sum__norm__bound,axiom,
    ! [S3: set_int,F2: fun_int_complex,K5: real] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),S3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(F2,X3))),K5)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups3049146728041665814omplex(F2,S3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_int(S3))),K5))) ) ).

% sum_norm_bound
tff(fact_5909_sum__norm__bound,axiom,
    ! [S3: set_list_nat,F2: fun_list_nat_complex,K5: real] :
      ( ! [X3: list_nat] :
          ( member_list_nat(X3,S3)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_list_nat_complex(F2,X3))),K5)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups6529277132148336714omplex(F2,S3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_list_nat(S3))),K5))) ) ).

% sum_norm_bound
tff(fact_5910_sum__norm__bound,axiom,
    ! [S3: set_set_nat,F2: fun_set_nat_complex,K5: real] :
      ( ! [X3: set_nat] :
          ( member_set_nat(X3,S3)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(F2,X3))),K5)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups8255218700646806128omplex(F2,S3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_set_nat(S3))),K5))) ) ).

% sum_norm_bound
tff(fact_5911_sum__norm__bound,axiom,
    ! [S3: set_nat,F2: fun_nat_real,K5: real] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),S3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F2,X3))),K5)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F2),S3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(finite_card_nat,S3))),K5))) ) ).

% sum_norm_bound
tff(fact_5912_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_rat,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% sum.ub_add_nat
tff(fact_5913_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_int,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups3539618377306564664at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),groups3539618377306564664at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% sum.ub_add_nat
tff(fact_5914_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_nat,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% sum.ub_add_nat
tff(fact_5915_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_real,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% sum.ub_add_nat
tff(fact_5916_even__two__times__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(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_5917_even__two__times__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(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_5918_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_5919_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5920_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_natural] :
      ( dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one)),A)
    <=> ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = zero_z2226904508553997617atural ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5921_power__mono__odd,axiom,
    ! [N: 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))),N))
     => ( 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(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ).

% power_mono_odd
tff(fact_5922_power__mono__odd,axiom,
    ! [N: 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))),N))
     => ( 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),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).

% power_mono_odd
tff(fact_5923_power__mono__odd,axiom,
    ! [N: 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))),N))
     => ( 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),N)),aa_nat_int(power_power_int(B),N))) ) ) ).

% power_mono_odd
tff(fact_5924_odd__pos,axiom,
    ! [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))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% odd_pos
tff(fact_5925_dvd__power__iff__le,axiom,
    ! [K2: nat,M2: nat,N: 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))),K2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% dvd_power_iff_le
tff(fact_5926_even__unset__bit__iff,axiom,
    ! [M2: 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,M2),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))
        | ( M2 = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_5927_even__unset__bit__iff,axiom,
    ! [M2: 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(M2),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))
        | ( M2 = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_5928_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_5929_even__set__bit__iff,axiom,
    ! [M2: 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,M2),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))
        & ( M2 != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5930_even__set__bit__iff,axiom,
    ! [M2: 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(M2),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))
        & ( M2 != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5931_even__flip__bit__iff,axiom,
    ! [M2: 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,M2),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))
        <=> ( M2 = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_5932_even__flip__bit__iff,axiom,
    ! [M2: 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(M2),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))
        <=> ( M2 = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_5933_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z2),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z2),M2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z2),aa_nat_rat(semiri681578069525770553at_rat,M2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).

% pochhammer_product
tff(fact_5934_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z2),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z2),M2)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(Z2),aa_nat_real(semiri5074537144036343181t_real,M2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).

% pochhammer_product
tff(fact_5935_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_int(comm_s4660882817536571857er_int(Z2),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z2),M2)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).

% pochhammer_product
tff(fact_5936_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),M2)),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,M2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).

% pochhammer_product
tff(fact_5937_oddE,axiom,
    ! [A: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ~ ! [B3: code_integer] : A != plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B3),one_one_Code_integer) ) ).

% oddE
tff(fact_5938_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_5939_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_5940_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_5941_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_5942_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(aa_int_fun_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(aa_int_fun_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_5943_mod2__eq__if,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one)),A)
       => ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = zero_z2226904508553997617atural ) )
      & ( ~ dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one)),A)
       => ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) = one_one_Code_natural ) ) ) ).

% mod2_eq_if
tff(fact_5944_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_5945_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_5946_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(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_5947_parity__cases,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one)),A)
       => ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) != zero_z2226904508553997617atural ) )
     => ~ ( ~ dvd_dvd_Code_natural(numera5444537566228673987atural(aa_num_num(bit0,one)),A)
         => ( modulo8411746178871703098atural(A,numera5444537566228673987atural(aa_num_num(bit0,one))) != one_one_Code_natural ) ) ) ).

% parity_cases
tff(fact_5948_zero__le__even__power,axiom,
    ! [N: 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))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% zero_le_even_power
tff(fact_5949_zero__le__even__power,axiom,
    ! [N: 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))),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% zero_le_even_power
tff(fact_5950_zero__le__even__power,axiom,
    ! [N: 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))),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_le_even_power
tff(fact_5951_zero__le__odd__power,axiom,
    ! [N: 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))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% zero_le_odd_power
tff(fact_5952_zero__le__odd__power,axiom,
    ! [N: 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))),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% zero_le_odd_power
tff(fact_5953_zero__le__odd__power,axiom,
    ! [N: 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))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% zero_le_odd_power
tff(fact_5954_zero__le__power__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_5955_zero__le__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_5956_zero__le__power__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_5957_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F2: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_hb(F2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(F2,M2)),aa_nat_complex(F2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_hb(F2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
tff(fact_5958_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F2: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( groups2906978787729119204at_rat(aTP_Lamm_hc(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F2,M2)),aa_nat_rat(F2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( groups2906978787729119204at_rat(aTP_Lamm_hc(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
tff(fact_5959_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F2: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( groups3539618377306564664at_int(aTP_Lamm_hd(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F2,M2)),aa_nat_int(F2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( groups3539618377306564664at_int(aTP_Lamm_hd(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = zero_zero_int ) ) ) ).

% sum_natinterval_diff
tff(fact_5960_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F2: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_he(F2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_real_real(minus_minus_real(aa_nat_real(F2,M2)),aa_nat_real(F2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_he(F2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = zero_zero_real ) ) ) ).

% sum_natinterval_diff
tff(fact_5961_sum__telescope_H_H,axiom,
    ! [M2: nat,N: nat,F2: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( groups2906978787729119204at_rat(aTP_Lamm_hf(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),N)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F2,N)),aa_nat_rat(F2,M2)) ) ) ).

% sum_telescope''
tff(fact_5962_sum__telescope_H_H,axiom,
    ! [M2: nat,N: nat,F2: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( groups3539618377306564664at_int(aTP_Lamm_hg(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F2,N)),aa_nat_int(F2,M2)) ) ) ).

% sum_telescope''
tff(fact_5963_sum__telescope_H_H,axiom,
    ! [M2: nat,N: nat,F2: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hh(F2)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),N)) = aa_real_real(minus_minus_real(aa_nat_real(F2,N)),aa_nat_real(F2,M2)) ) ) ).

% sum_telescope''
tff(fact_5964_even__set__encode__iff,axiom,
    ! [A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( 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,A4)))
      <=> ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) ) ) ).

% even_set_encode_iff
tff(fact_5965_zero__less__power__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
    <=> ( ( N = 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))),N))
          & ( 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))),N))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_5966_zero__less__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)))
    <=> ( ( N = 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))),N))
          & ( 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))),N))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_5967_zero__less__power__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
    <=> ( ( N = 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))),N))
          & ( 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))),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_5968_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(one_one_Code_integer,X)),groups7501900531339628137nteger(power_8256067586552552935nteger(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),M2),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_5969_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M2)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_5970_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),M2)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_5971_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( 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(M2),N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),M2)),aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_5972_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) = aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_5973_sum_Oin__pairs,axiom,
    ! [G2: fun_nat_rat,M2: nat,N: nat] : groups2906978787729119204at_rat(G2,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))),M2)),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)))) = groups2906978787729119204at_rat(aTP_Lamm_hi(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).

% sum.in_pairs
tff(fact_5974_sum_Oin__pairs,axiom,
    ! [G2: fun_nat_int,M2: nat,N: nat] : groups3539618377306564664at_int(G2,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))),M2)),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)))) = groups3539618377306564664at_int(aTP_Lamm_hj(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).

% sum.in_pairs
tff(fact_5975_sum_Oin__pairs,axiom,
    ! [G2: fun_nat_nat,M2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G2),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))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hk(G2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).

% sum.in_pairs
tff(fact_5976_sum_Oin__pairs,axiom,
    ! [G2: fun_nat_real,M2: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G2),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))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hl(G2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).

% sum.in_pairs
tff(fact_5977_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M2),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% even_mask_div_iff'
tff(fact_5978_even__mask__div__iff_H,axiom,
    ! [M2: nat,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))),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))),M2)),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))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% even_mask_div_iff'
tff(fact_5979_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: 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(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))),M2)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% even_mask_div_iff'
tff(fact_5980_power__le__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),zero_zero_real))
    <=> ( 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(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & 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))),N))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5981_power__le__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),zero_zero_rat))
    <=> ( 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(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & 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))),N))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5982_power__le__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),zero_zero_int))
    <=> ( 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(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & 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))),N))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5983_mask__eq__sum__exp__nat,axiom,
    ! [N: 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))),N)),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_hm,N))) ).

% mask_eq_sum_exp_nat
tff(fact_5984_gauss__sum__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_5985_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),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,N),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_5986_even__mask__div__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M2),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) = zero_z3403309356797280102nteger )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% even_mask_div_iff
tff(fact_5987_even__mask__div__iff,axiom,
    ! [M2: nat,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))),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))),M2)),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))),N))))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% even_mask_div_iff
tff(fact_5988_even__mask__div__iff,axiom,
    ! [M2: nat,N: 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(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))),M2)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% even_mask_div_iff
tff(fact_5989_double__gauss__sum,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups7501900531339628137nteger(semiri4939895301339042750nteger,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)) ).

% double_gauss_sum
tff(fact_5990_double__gauss__sum,axiom,
    ! [N: nat] : aa_rat_rat(aa_rat_fun_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),N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ).

% double_gauss_sum
tff(fact_5991_double__gauss__sum,axiom,
    ! [N: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),one_on7984719198319812577d_enat)) ).

% double_gauss_sum
tff(fact_5992_double__gauss__sum,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(semiri8010041392384452111omplex),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).

% double_gauss_sum
tff(fact_5993_double__gauss__sum,axiom,
    ! [N: 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),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum
tff(fact_5994_double__gauss__sum,axiom,
    ! [N: 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),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ).

% double_gauss_sum
tff(fact_5995_double__gauss__sum,axiom,
    ! [N: 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),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)) ).

% double_gauss_sum
tff(fact_5996_double__arith__series,axiom,
    ! [A: code_integer,D: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_hn(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),D))) ).

% double_arith_series
tff(fact_5997_double__arith__series,axiom,
    ! [A: rat,D: rat,N: nat] : aa_rat_rat(aa_rat_fun_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_ho(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),D))) ).

% double_arith_series
tff(fact_5998_double__arith__series,axiom,
    ! [A: extended_enat,D: extended_enat,N: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_hp(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = times_7803423173614009249d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),one_on7984719198319812577d_enat),plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),A),times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),D))) ).

% double_arith_series
tff(fact_5999_double__arith__series,axiom,
    ! [A: complex,D: complex,N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_hq(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),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,N)),D))) ).

% double_arith_series
tff(fact_6000_double__arith__series,axiom,
    ! [A: int,D: int,N: 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_hr(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = 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,N)),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,N)),D))) ).

% double_arith_series
tff(fact_6001_double__arith__series,axiom,
    ! [A: nat,D: nat,N: 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_hs(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = 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,N)),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,N)),D))) ).

% double_arith_series
tff(fact_6002_double__arith__series,axiom,
    ! [A: real,D: real,N: 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_ht(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_real_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,N)),D))) ).

% double_arith_series
tff(fact_6003_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hu(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = 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,N)),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,N),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_6004_Sum__Icc__nat,axiom,
    ! [M2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = 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,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_6005_Bernoulli__inequality__even,axiom,
    ! [N: 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))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(one_one_real),X)),N))) ) ).

% Bernoulli_inequality_even
tff(fact_6006_gauss__sum,axiom,
    ! [N: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_6007_gauss__sum,axiom,
    ! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_6008_gauss__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = 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,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_6009_arith__series,axiom,
    ! [A: code_integer,D: code_integer,N: nat] : groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_hn(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),D))),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% arith_series
tff(fact_6010_arith__series,axiom,
    ! [A: int,D: int,N: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_hr(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),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,N)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_6011_arith__series,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hs(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = 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,N)),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,N)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_6012_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups7501900531339628137nteger(semiri4939895301339042750nteger,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6013_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_rat_rat(aa_rat_fun_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)),N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6014_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : 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)),N))) = times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N),one_on7984719198319812577d_enat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6015_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(semiri8010041392384452111omplex),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6016_double__gauss__sum__from__Suc__0,axiom,
    ! [N: 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)),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6017_double__gauss__sum__from__Suc__0,axiom,
    ! [N: 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)),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6018_double__gauss__sum__from__Suc__0,axiom,
    ! [N: 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)),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6019_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M2: nat,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))),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))),M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
          & 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,N),M2))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6020_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M2: nat,N: 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(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))),M2))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
          & 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(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,N),M2))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6021_sum__gp__offset,axiom,
    ! [X: complex,M2: nat,N: nat] :
      ( ( ( X = one_one_complex )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex) ) )
      & ( ( X != one_one_complex )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M2)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp_offset
tff(fact_6022_sum__gp__offset,axiom,
    ! [X: rat,M2: nat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),M2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)) ) ) ) ).

% sum_gp_offset
tff(fact_6023_sum__gp__offset,axiom,
    ! [X: real,M2: nat,N: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real) ) )
      & ( ( X != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M2)),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))))),aa_real_real(minus_minus_real(one_one_real),X)) ) ) ) ).

% sum_gp_offset
tff(fact_6024_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va3: 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,Va3))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va3))) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
      & ( ~ 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,Va3))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va3))) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_6025_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_6026_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_6027_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = 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,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_6028_power__half__series,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_hv),one_one_real)) ).

% power_half_series
tff(fact_6029_sums__zero,axiom,
    pp(aa_complex_bool(sums_complex(aTP_Lamm_hw),zero_zero_complex)) ).

% sums_zero
tff(fact_6030_sums__zero,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_dd),zero_zero_real)) ).

% sums_zero
tff(fact_6031_sums__zero,axiom,
    pp(aa_nat_bool(sums_nat(aTP_Lamm_al),zero_zero_nat)) ).

% sums_zero
tff(fact_6032_sums__zero,axiom,
    pp(aa_int_bool(sums_int(aTP_Lamm_hx),zero_zero_int)) ).

% sums_zero
tff(fact_6033_powser__sums__if,axiom,
    ! [M2: nat,Z2: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_hy(M2),Z2)),aa_nat_complex(power_power_complex(Z2),M2))) ).

% powser_sums_if
tff(fact_6034_powser__sums__if,axiom,
    ! [M2: nat,Z2: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_hz(M2),Z2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),M2))) ).

% powser_sums_if
tff(fact_6035_powser__sums__if,axiom,
    ! [M2: nat,Z2: int] : pp(aa_int_bool(sums_int(aa_int_fun_nat_int(aTP_Lamm_ia(M2),Z2)),aa_nat_int(power_power_int(Z2),M2))) ).

% powser_sums_if
tff(fact_6036_sums__If__finite__set,axiom,
    ! [A4: set_nat,F2: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ib(A4),F2)),aa_set_nat_complex(groups2073611262835488442omplex(F2),A4))) ) ).

% sums_If_finite_set
tff(fact_6037_sums__If__finite__set,axiom,
    ! [A4: set_nat,F2: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_ic(A4),F2)),groups3539618377306564664at_int(F2,A4))) ) ).

% sums_If_finite_set
tff(fact_6038_sums__If__finite__set,axiom,
    ! [A4: set_nat,F2: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_id(A4),F2)),aa_set_nat_nat(groups3542108847815614940at_nat(F2),A4))) ) ).

% sums_If_finite_set
tff(fact_6039_sums__If__finite__set,axiom,
    ! [A4: set_nat,F2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ie(A4),F2)),aa_set_nat_real(groups6591440286371151544t_real(F2),A4))) ) ).

% sums_If_finite_set
tff(fact_6040_sums__If__finite,axiom,
    ! [P2: fun_nat_bool,F2: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P2)))
     => pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_if(P2),F2)),aa_set_nat_complex(groups2073611262835488442omplex(F2),collect_nat(P2)))) ) ).

% sums_If_finite
tff(fact_6041_sums__If__finite,axiom,
    ! [P2: fun_nat_bool,F2: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P2)))
     => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_ig(P2),F2)),groups3539618377306564664at_int(F2,collect_nat(P2)))) ) ).

% sums_If_finite
tff(fact_6042_sums__If__finite,axiom,
    ! [P2: fun_nat_bool,F2: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P2)))
     => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ih(P2),F2)),aa_set_nat_nat(groups3542108847815614940at_nat(F2),collect_nat(P2)))) ) ).

% sums_If_finite
tff(fact_6043_sums__If__finite,axiom,
    ! [P2: fun_nat_bool,F2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P2)))
     => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ii(P2),F2)),aa_set_nat_real(groups6591440286371151544t_real(F2),collect_nat(P2)))) ) ).

% sums_If_finite
tff(fact_6044_sums__finite,axiom,
    ! [N5: set_nat,F2: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N3: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N3),N5))
           => ( aa_nat_complex(F2,N3) = zero_zero_complex ) )
       => pp(aa_complex_bool(sums_complex(F2),aa_set_nat_complex(groups2073611262835488442omplex(F2),N5))) ) ) ).

% sums_finite
tff(fact_6045_sums__finite,axiom,
    ! [N5: set_nat,F2: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N3: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N3),N5))
           => ( aa_nat_int(F2,N3) = zero_zero_int ) )
       => pp(aa_int_bool(sums_int(F2),groups3539618377306564664at_int(F2,N5))) ) ) ).

% sums_finite
tff(fact_6046_sums__finite,axiom,
    ! [N5: set_nat,F2: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N3: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N3),N5))
           => ( aa_nat_nat(F2,N3) = zero_zero_nat ) )
       => pp(aa_nat_bool(sums_nat(F2),aa_set_nat_nat(groups3542108847815614940at_nat(F2),N5))) ) ) ).

% sums_finite
tff(fact_6047_sums__finite,axiom,
    ! [N5: set_nat,F2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N3: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N3),N5))
           => ( aa_nat_real(F2,N3) = zero_zero_real ) )
       => pp(aa_real_bool(sums_real(F2),aa_set_nat_real(groups6591440286371151544t_real(F2),N5))) ) ) ).

% sums_finite
tff(fact_6048_sums__zero__iff__shift,axiom,
    ! [N: nat,F2: fun_nat_complex,S2: complex] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => ( aa_nat_complex(F2,I3) = zero_zero_complex ) )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ij(N),F2)),S2))
      <=> pp(aa_complex_bool(sums_complex(F2),S2)) ) ) ).

% sums_zero_iff_shift
tff(fact_6049_sums__zero__iff__shift,axiom,
    ! [N: nat,F2: fun_nat_real,S2: real] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => ( aa_nat_real(F2,I3) = zero_zero_real ) )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ik(N),F2)),S2))
      <=> pp(aa_real_bool(sums_real(F2),S2)) ) ) ).

% sums_zero_iff_shift
tff(fact_6050_zdvd__mono,axiom,
    ! [K2: int,M2: int,T: int] :
      ( ( K2 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M2),T))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),M2)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),T))) ) ) ).

% zdvd_mono
tff(fact_6051_zdvd__mult__cancel,axiom,
    ! [K2: int,M2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),M2)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),N)))
     => ( ( K2 != zero_zero_int )
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M2),N)) ) ) ).

% zdvd_mult_cancel
tff(fact_6052_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C2: 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,C2),D))),T))) ) ) ).

% zdvd_period
tff(fact_6053_zdvd__reduce,axiom,
    ! [K2: int,N: int,M2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),aa_int_int(aa_int_fun_int_int(times_times_int,K2),M2))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K2),N)) ) ).

% zdvd_reduce
tff(fact_6054_sum__subtractf__nat,axiom,
    ! [A4: set_complex,G2: fun_complex_nat,F2: fun_complex_nat] :
      ( ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(G2,X3)),aa_complex_nat(F2,X3))) )
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_il(G2),F2),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F2,A4)),groups5693394587270226106ex_nat(G2,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_6055_sum__subtractf__nat,axiom,
    ! [A4: set_real,G2: fun_real_nat,F2: fun_real_nat] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(G2,X3)),aa_real_nat(F2,X3))) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_im(G2),F2),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F2,A4)),groups1935376822645274424al_nat(G2,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_6056_sum__subtractf__nat,axiom,
    ! [A4: set_set_nat,G2: fun_set_nat_nat,F2: fun_set_nat_nat] :
      ( ! [X3: set_nat] :
          ( member_set_nat(X3,A4)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(G2,X3)),aa_set_nat_nat(F2,X3))) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_in(G2),F2),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F2,A4)),groups8294997508430121362at_nat(G2,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_6057_sum__subtractf__nat,axiom,
    ! [A4: set_int,G2: fun_int_nat,F2: fun_int_nat] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(G2,X3)),aa_int_nat(F2,X3))) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_io(G2),F2),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F2,A4)),groups4541462559716669496nt_nat(G2,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_6058_sum__subtractf__nat,axiom,
    ! [A4: set_nat,G2: fun_nat_nat,F2: fun_nat_nat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(G2,X3)),aa_nat_nat(F2,X3))) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ip(G2),F2)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F2),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(G2),A4)) ) ) ).

% sum_subtractf_nat
tff(fact_6059_sum__eq__Suc0__iff,axiom,
    ! [A4: set_int,F2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( groups4541462559716669496nt_nat(F2,A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
            & ( aa_int_nat(F2,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),A4))
               => ( ( X4 != Xa3 )
                 => ( aa_int_nat(F2,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6060_sum__eq__Suc0__iff,axiom,
    ! [A4: set_complex,F2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( groups5693394587270226106ex_nat(F2,A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A4))
            & ( aa_complex_nat(F2,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa3),A4))
               => ( ( X4 != Xa3 )
                 => ( aa_complex_nat(F2,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6061_sum__eq__Suc0__iff,axiom,
    ! [A4: set_nat,F2: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F2),A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A4))
            & ( aa_nat_nat(F2,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa3),A4))
               => ( ( X4 != Xa3 )
                 => ( aa_nat_nat(F2,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6062_sum__SucD,axiom,
    ! [F2: fun_nat_nat,A4: set_nat,N: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F2),A4) = aa_nat_nat(suc,N) )
     => ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A4))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F2,X3))) ) ) ).

% sum_SucD
tff(fact_6063_sum__eq__1__iff,axiom,
    ! [A4: set_int,F2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( groups4541462559716669496nt_nat(F2,A4) = one_one_nat )
      <=> ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
            & ( aa_int_nat(F2,X4) = one_one_nat )
            & ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),A4))
               => ( ( X4 != Xa3 )
                 => ( aa_int_nat(F2,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6064_sum__eq__1__iff,axiom,
    ! [A4: set_complex,F2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( groups5693394587270226106ex_nat(F2,A4) = one_one_nat )
      <=> ? [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A4))
            & ( aa_complex_nat(F2,X4) = one_one_nat )
            & ! [Xa3: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa3),A4))
               => ( ( X4 != Xa3 )
                 => ( aa_complex_nat(F2,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6065_sum__eq__1__iff,axiom,
    ! [A4: set_nat,F2: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F2),A4) = one_one_nat )
      <=> ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A4))
            & ( aa_nat_nat(F2,X4) = one_one_nat )
            & ! [Xa3: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa3),A4))
               => ( ( X4 != Xa3 )
                 => ( aa_nat_nat(F2,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6066_sum__Suc,axiom,
    ! [F2: fun_complex_nat,A4: set_complex] : groups5693394587270226106ex_nat(aTP_Lamm_iq(F2),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(F2,A4)),finite_card_complex(A4)) ).

% sum_Suc
tff(fact_6067_sum__Suc,axiom,
    ! [F2: fun_int_nat,A4: set_int] : groups4541462559716669496nt_nat(aTP_Lamm_ir(F2),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(F2,A4)),finite_card_int(A4)) ).

% sum_Suc
tff(fact_6068_sum__Suc,axiom,
    ! [F2: fun_list_nat_nat,A4: set_list_nat] : groups4396056296759096172at_nat(aTP_Lamm_is(F2),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4396056296759096172at_nat(F2,A4)),finite_card_list_nat(A4)) ).

% sum_Suc
tff(fact_6069_sum__Suc,axiom,
    ! [F2: fun_set_nat_nat,A4: set_set_nat] : groups8294997508430121362at_nat(aTP_Lamm_it(F2),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups8294997508430121362at_nat(F2,A4)),finite_card_set_nat(A4)) ).

% sum_Suc
tff(fact_6070_sum__Suc,axiom,
    ! [F2: fun_nat_nat,A4: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_iu(F2)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F2),A4)),aa_set_nat_nat(finite_card_nat,A4)) ).

% sum_Suc
tff(fact_6071_sum__multicount,axiom,
    ! [S3: set_real,T4: set_real,R2: fun_re1255401033344146673l_bool,K2: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),T4))
             => ( finite_card_real(collect_real(aa_rea7809428808904475482l_bool(aa_fun1425803159125577980l_bool(aTP_Lamm_iv(S3),R2),X3))) = K2 ) )
         => ( groups1935376822645274424al_nat(aa_fun3074920444450379950al_nat(aTP_Lamm_ix(T4),R2),S3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),finite_card_real(T4)) ) ) ) ) ).

% sum_multicount
tff(fact_6072_sum__multicount,axiom,
    ! [S3: set_real,T4: set_nat,R2: fun_re9121486324101951821t_bool,K2: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T4))
       => ( ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),T4))
             => ( finite_card_real(collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_iy(S3),R2),X3))) = K2 ) )
         => ( groups1935376822645274424al_nat(aa_fun5804151645385689226al_nat(aTP_Lamm_ja(T4),R2),S3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_set_nat_nat(finite_card_nat,T4)) ) ) ) ) ).

% sum_multicount
tff(fact_6073_sum__multicount,axiom,
    ! [S3: set_real,T4: set_int,R2: fun_re8661203372455305201t_bool,K2: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),T4))
             => ( finite_card_real(collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_jb(S3),R2),X3))) = K2 ) )
         => ( groups1935376822645274424al_nat(aa_fun4707068664741690670al_nat(aTP_Lamm_jd(T4),R2),S3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),finite_card_int(T4)) ) ) ) ) ).

% sum_multicount
tff(fact_6074_sum__multicount,axiom,
    ! [S3: set_real,T4: set_complex,R2: fun_re3010880487650781423x_bool,K2: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),T4))
             => ( finite_card_real(collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_je(S3),R2),X3))) = K2 ) )
         => ( groups1935376822645274424al_nat(aa_fun5167920629197714732al_nat(aTP_Lamm_jg(T4),R2),S3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),finite_card_complex(T4)) ) ) ) ) ).

% sum_multicount
tff(fact_6075_sum__multicount,axiom,
    ! [S3: set_int,T4: set_real,R2: fun_in3970322703178350321l_bool,K2: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),T4))
             => ( finite_card_int(collect_int(aa_real_fun_int_bool(aa_fun1009312342505620988t_bool(aTP_Lamm_jh(S3),R2),X3))) = K2 ) )
         => ( groups4541462559716669496nt_nat(aa_fun91123856807183534nt_nat(aTP_Lamm_jj(T4),R2),S3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),finite_card_real(T4)) ) ) ) ) ).

% sum_multicount
tff(fact_6076_sum__multicount,axiom,
    ! [S3: set_int,T4: set_nat,R2: fun_int_fun_nat_bool,K2: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T4))
       => ( ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),T4))
             => ( finite_card_int(collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_jk(S3),R2),X3))) = K2 ) )
         => ( groups4541462559716669496nt_nat(aa_fun5132377469993263754nt_nat(aTP_Lamm_jm(T4),R2),S3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_set_nat_nat(finite_card_nat,T4)) ) ) ) ) ).

% sum_multicount
tff(fact_6077_sum__multicount,axiom,
    ! [S3: set_int,T4: set_int,R2: fun_int_fun_int_bool,K2: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),T4))
             => ( finite_card_int(collect_int(aa_int_fun_int_bool(aa_fun8168818243683596540t_bool(aTP_Lamm_jn(S3),R2),X3))) = K2 ) )
         => ( groups4541462559716669496nt_nat(aa_fun8170219554457397550nt_nat(aTP_Lamm_jp(T4),R2),S3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),finite_card_int(T4)) ) ) ) ) ).

% sum_multicount
tff(fact_6078_sum__multicount,axiom,
    ! [S3: set_int,T4: set_complex,R2: fun_in5750378413879446255x_bool,K2: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),T4))
             => ( finite_card_int(collect_int(aa_com2211550117870696024t_bool(aa_fun3087709329828230012t_bool(aTP_Lamm_jq(S3),R2),X3))) = K2 ) )
         => ( groups4541462559716669496nt_nat(aa_fun8429035501195100204nt_nat(aTP_Lamm_js(T4),R2),S3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),finite_card_complex(T4)) ) ) ) ) ).

% sum_multicount
tff(fact_6079_sum__multicount,axiom,
    ! [S3: set_complex,T4: set_real,R2: fun_co2286642472385812463l_bool,K2: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [X3: real] :
              ( pp(aa_set_real_bool(member_real(X3),T4))
             => ( finite_card_complex(collect_complex(aa_rea5887071448072802392x_bool(aa_fun5411146308205027452x_bool(aTP_Lamm_jt(S3),R2),X3))) = K2 ) )
         => ( groups5693394587270226106ex_nat(aa_fun2409747231911459374ex_nat(aTP_Lamm_jv(T4),R2),S3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),finite_card_real(T4)) ) ) ) ) ).

% sum_multicount
tff(fact_6080_sum__multicount,axiom,
    ! [S3: set_complex,T4: set_nat,R2: fun_co2841405903858895691t_bool,K2: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T4))
       => ( ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),T4))
             => ( finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_jw(S3),R2),X3))) = K2 ) )
         => ( groups5693394587270226106ex_nat(aa_fun9050545535516344074ex_nat(aTP_Lamm_jy(T4),R2),S3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_set_nat_nat(finite_card_nat,T4)) ) ) ) ) ).

% sum_multicount
tff(fact_6081_mod__int__pos__iff,axiom,
    ! [K2: int,L: 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(modulo_modulo_int,K2),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K2))
        | ( ( L = zero_zero_int )
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2)) )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L)) ) ) ).

% mod_int_pos_iff
tff(fact_6082_sum__count__set,axiom,
    ! [Xs: list_complex,X6: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Xs)),X6))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,X6))
       => ( groups5693394587270226106ex_nat(count_list_complex(Xs),X6) = size_s3451745648224563538omplex(Xs) ) ) ) ).

% sum_count_set
tff(fact_6083_sum__count__set,axiom,
    ! [Xs: list_VEBT_VEBT,X6: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),X6)
     => ( finite5795047828879050333T_VEBT(X6)
       => ( groups771621172384141258BT_nat(count_list_VEBT_VEBT(Xs),X6) = size_s6755466524823107622T_VEBT(Xs) ) ) ) ).

% sum_count_set
tff(fact_6084_sum__count__set,axiom,
    ! [Xs: list_bool,X6: set_bool] :
      ( ord_less_eq_set_bool(set_bool2(Xs),X6)
     => ( finite_finite_bool(X6)
       => ( groups5768501273728744877ol_nat(count_list_bool(Xs),X6) = size_size_list_bool(Xs) ) ) ) ).

% sum_count_set
tff(fact_6085_sum__count__set,axiom,
    ! [Xs: list_int,X6: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),X6))
     => ( pp(aa_set_int_bool(finite_finite_int,X6))
       => ( groups4541462559716669496nt_nat(count_list_int(Xs),X6) = size_size_list_int(Xs) ) ) ) ).

% sum_count_set
tff(fact_6086_sum__count__set,axiom,
    ! [Xs: list_nat,X6: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Xs)),X6))
     => ( pp(aa_set_nat_bool(finite_finite_nat,X6))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(count_list_nat(Xs)),X6) = aa_list_nat_nat(size_size_list_nat,Xs) ) ) ) ).

% sum_count_set
tff(fact_6087_sums__le,axiom,
    ! [F2: fun_nat_real,G2: fun_nat_real,S2: real,T: real] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F2,N3)),aa_nat_real(G2,N3)))
     => ( pp(aa_real_bool(sums_real(F2),S2))
       => ( pp(aa_real_bool(sums_real(G2),T))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,S2),T)) ) ) ) ).

% sums_le
tff(fact_6088_sums__le,axiom,
    ! [F2: fun_nat_nat,G2: fun_nat_nat,S2: nat,T: nat] :
      ( ! [N3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F2,N3)),aa_nat_nat(G2,N3)))
     => ( pp(aa_nat_bool(sums_nat(F2),S2))
       => ( pp(aa_nat_bool(sums_nat(G2),T))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,S2),T)) ) ) ) ).

% sums_le
tff(fact_6089_sums__le,axiom,
    ! [F2: fun_nat_int,G2: fun_nat_int,S2: int,T: int] :
      ( ! [N3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F2,N3)),aa_nat_int(G2,N3)))
     => ( pp(aa_int_bool(sums_int(F2),S2))
       => ( pp(aa_int_bool(sums_int(G2),T))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,S2),T)) ) ) ) ).

% sums_le
tff(fact_6090_sums__0,axiom,
    ! [F2: fun_nat_complex] :
      ( ! [N3: nat] : aa_nat_complex(F2,N3) = zero_zero_complex
     => pp(aa_complex_bool(sums_complex(F2),zero_zero_complex)) ) ).

% sums_0
tff(fact_6091_sums__0,axiom,
    ! [F2: fun_nat_real] :
      ( ! [N3: nat] : aa_nat_real(F2,N3) = zero_zero_real
     => pp(aa_real_bool(sums_real(F2),zero_zero_real)) ) ).

% sums_0
tff(fact_6092_sums__0,axiom,
    ! [F2: fun_nat_nat] :
      ( ! [N3: nat] : aa_nat_nat(F2,N3) = zero_zero_nat
     => pp(aa_nat_bool(sums_nat(F2),zero_zero_nat)) ) ).

% sums_0
tff(fact_6093_sums__0,axiom,
    ! [F2: fun_nat_int] :
      ( ! [N3: nat] : aa_nat_int(F2,N3) = zero_zero_int
     => pp(aa_int_bool(sums_int(F2),zero_zero_int)) ) ).

% sums_0
tff(fact_6094_sums__single,axiom,
    ! [I2: nat,F2: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_df(I2),F2)),aa_nat_complex(F2,I2))) ).

% sums_single
tff(fact_6095_sums__single,axiom,
    ! [I2: nat,F2: fun_nat_real] : pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_jz(I2),F2)),aa_nat_real(F2,I2))) ).

% sums_single
tff(fact_6096_sums__single,axiom,
    ! [I2: nat,F2: fun_nat_nat] : pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ka(I2),F2)),aa_nat_nat(F2,I2))) ).

% sums_single
tff(fact_6097_sums__single,axiom,
    ! [I2: nat,F2: fun_nat_int] : pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_kb(I2),F2)),aa_nat_int(F2,I2))) ).

% sums_single
tff(fact_6098_sums__mult2,axiom,
    ! [F2: fun_nat_real,A: real,C2: real] :
      ( pp(aa_real_bool(sums_real(F2),A))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_et(F2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))) ) ).

% sums_mult2
tff(fact_6099_sums__mult,axiom,
    ! [F2: fun_nat_real,A: real,C2: real] :
      ( pp(aa_real_bool(sums_real(F2),A))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_kc(F2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A))) ) ).

% sums_mult
tff(fact_6100_Sum__Icc__int,axiom,
    ! [M2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M2),N))
     => ( groups4538972089207619220nt_int(aTP_Lamm_as,set_or1266510415728281911st_int(M2,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M2),aa_int_int(aa_int_fun_int_int(minus_minus_int,M2),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_6101_sums__mult__iff,axiom,
    ! [C2: complex,F2: fun_nat_complex,D: complex] :
      ( ( C2 != zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_kd(C2),F2)),aa_complex_complex(times_times_complex(C2),D)))
      <=> pp(aa_complex_bool(sums_complex(F2),D)) ) ) ).

% sums_mult_iff
tff(fact_6102_sums__mult__iff,axiom,
    ! [C2: real,F2: fun_nat_real,D: real] :
      ( ( C2 != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_eq(C2),F2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),D)))
      <=> pp(aa_real_bool(sums_real(F2),D)) ) ) ).

% sums_mult_iff
tff(fact_6103_sums__mult2__iff,axiom,
    ! [C2: complex,F2: fun_nat_complex,D: complex] :
      ( ( C2 != zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ke(C2),F2)),aa_complex_complex(times_times_complex(D),C2)))
      <=> pp(aa_complex_bool(sums_complex(F2),D)) ) ) ).

% sums_mult2_iff
tff(fact_6104_sums__mult2__iff,axiom,
    ! [C2: real,F2: fun_nat_real,D: real] :
      ( ( C2 != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_kf(C2),F2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,D),C2)))
      <=> pp(aa_real_bool(sums_real(F2),D)) ) ) ).

% sums_mult2_iff
tff(fact_6105_sums__mult__D,axiom,
    ! [C2: complex,F2: fun_nat_complex,A: complex] :
      ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_kd(C2),F2)),A))
     => ( ( C2 != zero_zero_complex )
       => pp(aa_complex_bool(sums_complex(F2),aa_complex_complex(divide1717551699836669952omplex(A),C2))) ) ) ).

% sums_mult_D
tff(fact_6106_sums__mult__D,axiom,
    ! [C2: real,F2: fun_nat_real,A: real] :
      ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_eq(C2),F2)),A))
     => ( ( C2 != zero_zero_real )
       => pp(aa_real_bool(sums_real(F2),aa_real_real(divide_divide_real(A),C2))) ) ) ).

% sums_mult_D
tff(fact_6107_sums__Suc__imp,axiom,
    ! [F2: fun_nat_complex,S2: complex] :
      ( ( aa_nat_complex(F2,zero_zero_nat) = zero_zero_complex )
     => ( pp(aa_complex_bool(sums_complex(aTP_Lamm_kg(F2)),S2))
       => pp(aa_complex_bool(sums_complex(F2),S2)) ) ) ).

% sums_Suc_imp
tff(fact_6108_sums__Suc__imp,axiom,
    ! [F2: fun_nat_real,S2: real] :
      ( ( aa_nat_real(F2,zero_zero_nat) = zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aTP_Lamm_gn(F2)),S2))
       => pp(aa_real_bool(sums_real(F2),S2)) ) ) ).

% sums_Suc_imp
tff(fact_6109_sums__Suc__iff,axiom,
    ! [F2: fun_nat_real,S2: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_gn(F2)),S2))
    <=> pp(aa_real_bool(sums_real(F2),aa_real_real(plus_plus_real(S2),aa_nat_real(F2,zero_zero_nat)))) ) ).

% sums_Suc_iff
tff(fact_6110_sums__Suc,axiom,
    ! [F2: fun_nat_real,L: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_gn(F2)),L))
     => pp(aa_real_bool(sums_real(F2),aa_real_real(plus_plus_real(L),aa_nat_real(F2,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_6111_sums__Suc,axiom,
    ! [F2: fun_nat_nat,L: nat] :
      ( pp(aa_nat_bool(sums_nat(aTP_Lamm_gm(F2)),L))
     => pp(aa_nat_bool(sums_nat(F2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),aa_nat_nat(F2,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_6112_sums__Suc,axiom,
    ! [F2: fun_nat_int,L: int] :
      ( pp(aa_int_bool(sums_int(aTP_Lamm_gx(F2)),L))
     => pp(aa_int_bool(sums_int(F2),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),aa_nat_int(F2,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_6113_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: complex,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z2),aa_nat_nat(suc,N))),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,N))) = aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_kh(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))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_6114_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z2),aa_nat_nat(suc,N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = groups73079841787564623at_rat(aTP_Lamm_ki(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))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_6115_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z2),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(Z2),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_kj,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))),N)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_6116_lemma__termdiff2,axiom,
    ! [H: complex,Z2: complex,N: 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)),N)),aa_nat_complex(power_power_complex(Z2),N))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(power_power_complex(Z2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_complex_complex(times_times_complex(H),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_kl(H),Z2),N)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6117_lemma__termdiff2,axiom,
    ! [H: rat,Z2: rat,N: nat] :
      ( ( H != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z2),H)),N)),aa_nat_rat(power_power_rat(Z2),N))),H)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(power_power_rat(Z2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_kn(H),Z2),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6118_lemma__termdiff2,axiom,
    ! [H: real,Z2: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Z2),H)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),N))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),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_kp(H),Z2),N)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6119_pochhammer__code,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N) = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N) = set_fo1084959871951514735nteger(aTP_Lamm_kq(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),one_one_Code_integer) ) ) ) ).

% pochhammer_code
tff(fact_6120_pochhammer__code,axiom,
    ! [N: nat,A: complex] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_kr(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_complex) ) ) ) ).

% pochhammer_code
tff(fact_6121_pochhammer__code,axiom,
    ! [N: nat,A: rat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_ks(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_rat) ) ) ) ).

% pochhammer_code
tff(fact_6122_pochhammer__code,axiom,
    ! [N: nat,A: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_kt(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_real) ) ) ) ).

% pochhammer_code
tff(fact_6123_pochhammer__code,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_ku(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_int) ) ) ) ).

% pochhammer_code
tff(fact_6124_pochhammer__code,axiom,
    ! [N: nat,A: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_kv(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_nat) ) ) ) ).

% pochhammer_code
tff(fact_6125_of__nat__code,axiom,
    ! [N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,N) = semiri4055485073559036834nteger(aTP_Lamm_kw,N,zero_z3403309356797280102nteger) ).

% of_nat_code
tff(fact_6126_of__nat__code,axiom,
    ! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,N) = aa_complex_complex(semiri2816024913162550771omplex(aTP_Lamm_kx,N),zero_zero_complex) ).

% of_nat_code
tff(fact_6127_of__nat__code,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,N) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_ky,N),zero_zero_rat) ).

% of_nat_code
tff(fact_6128_of__nat__code,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_kz),N),zero_zero_real) ).

% of_nat_code
tff(fact_6129_of__nat__code,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_int_int(semiri8420488043553186161ux_int(aTP_Lamm_la,N),zero_zero_int) ).

% of_nat_code
tff(fact_6130_of__nat__code,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_lb),N),zero_zero_nat) ).

% of_nat_code
tff(fact_6131_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),N)),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,N)))),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))),N)),N)))) ) ).

% central_binomial_lower_bound
tff(fact_6132_concat__bit__Suc,axiom,
    ! [N: nat,K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(suc,N)),K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_6133_dbl__simps_I3_J,axiom,
    neg_nu8804712462038260780nteger(one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_6134_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_6135_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_6136_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_6137_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_6138_lessThan__iff,axiom,
    ! [I2: set_nat,K2: set_nat] :
      ( member_set_nat(I2,set_or890127255671739683et_nat(K2))
    <=> pp(aa_set_nat_bool(ord_less_set_nat(I2),K2)) ) ).

% lessThan_iff
tff(fact_6139_lessThan__iff,axiom,
    ! [I2: real,K2: real] :
      ( pp(aa_set_real_bool(member_real(I2),set_or5984915006950818249n_real(K2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I2),K2)) ) ).

% lessThan_iff
tff(fact_6140_lessThan__iff,axiom,
    ! [I2: rat,K2: rat] :
      ( member_rat(I2,set_ord_lessThan_rat(K2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I2),K2)) ) ).

% lessThan_iff
tff(fact_6141_lessThan__iff,axiom,
    ! [I2: num,K2: num] :
      ( member_num(I2,set_ord_lessThan_num(K2))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,I2),K2)) ) ).

% lessThan_iff
tff(fact_6142_lessThan__iff,axiom,
    ! [I2: int,K2: int] :
      ( pp(aa_set_int_bool(member_int(I2),set_ord_lessThan_int(K2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),K2)) ) ).

% lessThan_iff
tff(fact_6143_lessThan__iff,axiom,
    ! [I2: nat,K2: nat] :
      ( pp(aa_set_nat_bool(member_nat(I2),aa_nat_set_nat(set_ord_lessThan_nat,K2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K2)) ) ).

% lessThan_iff
tff(fact_6144_finite__lessThan,axiom,
    ! [K2: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_ord_lessThan_nat,K2))) ).

% finite_lessThan
tff(fact_6145_concat__bit__0,axiom,
    ! [K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(zero_zero_nat),K2),L) = L ).

% concat_bit_0
tff(fact_6146_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_6147_dbl__simps_I2_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,zero_zero_complex) = zero_zero_complex ).

% dbl_simps(2)
tff(fact_6148_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_6149_dbl__simps_I2_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_6150_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_6151_prod__zero__iff,axiom,
    ! [A4: set_nat,F2: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( aa_set_nat_complex(groups6464643781859351333omplex(F2),A4) = zero_zero_complex )
      <=> ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A4))
            & ( aa_nat_complex(F2,X4) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_6152_prod__zero__iff,axiom,
    ! [A4: set_int,F2: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( groups7440179247065528705omplex(F2,A4) = zero_zero_complex )
      <=> ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
            & ( aa_int_complex(F2,X4) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_6153_prod__zero__iff,axiom,
    ! [A4: set_complex,F2: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( groups3708469109370488835omplex(F2,A4) = zero_zero_complex )
      <=> ? [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A4))
            & ( aa_complex_complex(F2,X4) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_6154_prod__zero__iff,axiom,
    ! [A4: set_nat,F2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( aa_set_nat_real(groups129246275422532515t_real(F2),A4) = zero_zero_real )
      <=> ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A4))
            & ( aa_nat_real(F2,X4) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_6155_prod__zero__iff,axiom,
    ! [A4: set_int,F2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( groups2316167850115554303t_real(F2,A4) = zero_zero_real )
      <=> ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
            & ( aa_int_real(F2,X4) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_6156_prod__zero__iff,axiom,
    ! [A4: set_complex,F2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( groups766887009212190081x_real(F2,A4) = zero_zero_real )
      <=> ? [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A4))
            & ( aa_complex_real(F2,X4) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_6157_prod__zero__iff,axiom,
    ! [A4: set_nat,F2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( groups73079841787564623at_rat(F2,A4) = zero_zero_rat )
      <=> ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A4))
            & ( aa_nat_rat(F2,X4) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_6158_prod__zero__iff,axiom,
    ! [A4: set_int,F2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( groups1072433553688619179nt_rat(F2,A4) = zero_zero_rat )
      <=> ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
            & ( aa_int_rat(F2,X4) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_6159_prod__zero__iff,axiom,
    ! [A4: set_complex,F2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( groups225925009352817453ex_rat(F2,A4) = zero_zero_rat )
      <=> ? [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A4))
            & ( aa_complex_rat(F2,X4) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_6160_prod__zero__iff,axiom,
    ! [A4: set_int,F2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( groups1707563613775114915nt_nat(F2,A4) = zero_zero_nat )
      <=> ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
            & ( aa_int_nat(F2,X4) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_6161_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_6162_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_6163_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_6164_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_6165_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_6166_dbl__simps_I5_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K2)) = numera6690914467698888265omplex(aa_num_num(bit0,K2)) ).

% dbl_simps(5)
tff(fact_6167_dbl__simps_I5_J,axiom,
    ! [K2: num] : aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K2)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,K2)) ).

% dbl_simps(5)
tff(fact_6168_dbl__simps_I5_J,axiom,
    ! [K2: num] : aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K2)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)) ).

% dbl_simps(5)
tff(fact_6169_prod_Oinsert,axiom,
    ! [A4: set_real,X: real,G2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ~ pp(aa_set_real_bool(member_real(X),A4))
       => ( groups1681761925125756287l_real(G2,insert_real(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G2,X)),groups1681761925125756287l_real(G2,A4)) ) ) ) ).

% prod.insert
tff(fact_6170_prod_Oinsert,axiom,
    ! [A4: set_nat,X: nat,G2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
       => ( aa_set_nat_real(groups129246275422532515t_real(G2),insert_nat(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G2,X)),aa_set_nat_real(groups129246275422532515t_real(G2),A4)) ) ) ) ).

% prod.insert
tff(fact_6171_prod_Oinsert,axiom,
    ! [A4: set_int,X: int,G2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ~ pp(aa_set_int_bool(member_int(X),A4))
       => ( groups2316167850115554303t_real(G2,insert_int(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G2,X)),groups2316167850115554303t_real(G2,A4)) ) ) ) ).

% prod.insert
tff(fact_6172_prod_Oinsert,axiom,
    ! [A4: set_complex,X: complex,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups766887009212190081x_real(G2,insert_complex(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G2,X)),groups766887009212190081x_real(G2,A4)) ) ) ) ).

% prod.insert
tff(fact_6173_prod_Oinsert,axiom,
    ! [A4: set_real,X: real,G2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ~ pp(aa_set_real_bool(member_real(X),A4))
       => ( groups4061424788464935467al_rat(G2,insert_real(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(G2,X)),groups4061424788464935467al_rat(G2,A4)) ) ) ) ).

% prod.insert
tff(fact_6174_prod_Oinsert,axiom,
    ! [A4: set_nat,X: nat,G2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
       => ( groups73079841787564623at_rat(G2,insert_nat(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G2,X)),groups73079841787564623at_rat(G2,A4)) ) ) ) ).

% prod.insert
tff(fact_6175_prod_Oinsert,axiom,
    ! [A4: set_int,X: int,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ~ pp(aa_set_int_bool(member_int(X),A4))
       => ( groups1072433553688619179nt_rat(G2,insert_int(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(G2,X)),groups1072433553688619179nt_rat(G2,A4)) ) ) ) ).

% prod.insert
tff(fact_6176_prod_Oinsert,axiom,
    ! [A4: set_complex,X: complex,G2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ~ pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups225925009352817453ex_rat(G2,insert_complex(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_complex_rat(G2,X)),groups225925009352817453ex_rat(G2,A4)) ) ) ) ).

% prod.insert
tff(fact_6177_prod_Oinsert,axiom,
    ! [A4: set_real,X: real,G2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ~ pp(aa_set_real_bool(member_real(X),A4))
       => ( groups4696554848551431203al_nat(G2,insert_real(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_real_nat(G2,X)),groups4696554848551431203al_nat(G2,A4)) ) ) ) ).

% prod.insert
tff(fact_6178_prod_Oinsert,axiom,
    ! [A4: set_int,X: int,G2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ~ pp(aa_set_int_bool(member_int(X),A4))
       => ( groups1707563613775114915nt_nat(G2,insert_int(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(G2,X)),groups1707563613775114915nt_nat(G2,A4)) ) ) ) ).

% prod.insert
tff(fact_6179_sum_OlessThan__Suc,axiom,
    ! [G2: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_rat(G2,N)) ).

% sum.lessThan_Suc
tff(fact_6180_sum_OlessThan__Suc,axiom,
    ! [G2: fun_nat_int,N: nat] : groups3539618377306564664at_int(G2,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G2,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_int(G2,N)) ).

% sum.lessThan_Suc
tff(fact_6181_sum_OlessThan__Suc,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_nat(G2,N)) ).

% sum.lessThan_Suc
tff(fact_6182_sum_OlessThan__Suc,axiom,
    ! [G2: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_real(G2,N)) ).

% sum.lessThan_Suc
tff(fact_6183_single__Diff__lessThan,axiom,
    ! [K2: int] : minus_minus_set_int(insert_int(K2,bot_bot_set_int),set_ord_lessThan_int(K2)) = insert_int(K2,bot_bot_set_int) ).

% single_Diff_lessThan
tff(fact_6184_single__Diff__lessThan,axiom,
    ! [K2: real] : minus_minus_set_real(insert_real(K2,bot_bot_set_real),set_or5984915006950818249n_real(K2)) = insert_real(K2,bot_bot_set_real) ).

% single_Diff_lessThan
tff(fact_6185_single__Diff__lessThan,axiom,
    ! [K2: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(K2,bot_bot_set_nat)),aa_nat_set_nat(set_ord_lessThan_nat,K2)) = insert_nat(K2,bot_bot_set_nat) ).

% single_Diff_lessThan
tff(fact_6186_prod_OlessThan__Suc,axiom,
    ! [G2: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_real(G2,N)) ).

% prod.lessThan_Suc
tff(fact_6187_prod_OlessThan__Suc,axiom,
    ! [G2: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G2,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G2,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_rat(G2,N)) ).

% prod.lessThan_Suc
tff(fact_6188_prod_OlessThan__Suc,axiom,
    ! [G2: fun_nat_int,N: nat] : groups705719431365010083at_int(G2,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G2,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_int(G2,N)) ).

% prod.lessThan_Suc
tff(fact_6189_prod_OlessThan__Suc,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_nat(G2,N)) ).

% prod.lessThan_Suc
tff(fact_6190_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: fun_nat_Code_integer] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups3455450783089532116nteger(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = one_one_Code_integer ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups3455450783089532116nteger(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3455450783089532116nteger(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_Code_integer(G2,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_6191_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_complex(groups6464643781859351333omplex(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = one_one_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_complex(groups6464643781859351333omplex(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_set_nat_complex(groups6464643781859351333omplex(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_complex(G2,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_6192_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = one_one_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_real(G2,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_6193_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups73079841787564623at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = one_one_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups73079841787564623at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_rat(G2,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_6194_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = one_one_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_int(G2,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_6195_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = one_one_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_nat(G2,aa_nat_nat(suc,N))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_6196_prod_Onat__diff__reindex,axiom,
    ! [G2: fun_nat_int,N: nat] : groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_lc(G2),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) = groups705719431365010083at_int(G2,aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nat_diff_reindex
tff(fact_6197_prod_Onat__diff__reindex,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ld(G2),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nat_diff_reindex
tff(fact_6198_lessThan__non__empty,axiom,
    ! [X: int] : set_ord_lessThan_int(X) != bot_bot_set_int ).

% lessThan_non_empty
tff(fact_6199_lessThan__non__empty,axiom,
    ! [X: real] : set_or5984915006950818249n_real(X) != bot_bot_set_real ).

% lessThan_non_empty
tff(fact_6200_infinite__Iio,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(finite_finite_int,set_ord_lessThan_int(A))) ).

% infinite_Iio
tff(fact_6201_prod_Odistrib,axiom,
    ! [G2: fun_nat_int,H: fun_nat_int,A4: set_nat] : groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_le(G2),H),A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G2,A4)),groups705719431365010083at_int(H,A4)) ).

% prod.distrib
tff(fact_6202_prod_Odistrib,axiom,
    ! [G2: fun_int_int,H: fun_int_int,A4: set_int] : groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_lf(G2),H),A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G2,A4)),groups1705073143266064639nt_int(H,A4)) ).

% prod.distrib
tff(fact_6203_prod_Odistrib,axiom,
    ! [G2: fun_nat_nat,H: fun_nat_nat,A4: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_lg(G2),H)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G2),A4)),aa_set_nat_nat(groups708209901874060359at_nat(H),A4)) ).

% prod.distrib
tff(fact_6204_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aTP_Lamm_lh(U)) ).

% lessThan_def
tff(fact_6205_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_li,U)) ).

% lessThan_def
tff(fact_6206_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_lj,U)) ).

% lessThan_def
tff(fact_6207_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_lk,U)) ).

% lessThan_def
tff(fact_6208_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_ll,U)) ).

% lessThan_def
tff(fact_6209_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hm,U)) ).

% lessThan_def
tff(fact_6210_prod_OlessThan__Suc__shift,axiom,
    ! [G2: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G2,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_gn(G2)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_6211_prod_OlessThan__Suc__shift,axiom,
    ! [G2: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G2,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G2,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_gw(G2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_6212_prod_OlessThan__Suc__shift,axiom,
    ! [G2: fun_nat_int,N: nat] : groups705719431365010083at_int(G2,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G2,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_gx(G2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_6213_prod_OlessThan__Suc__shift,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G2,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_gm(G2)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.lessThan_Suc_shift
tff(fact_6214_prod_OatLeast1__atMost__eq,axiom,
    ! [G2: fun_nat_int,N: nat] : groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = groups705719431365010083at_int(aTP_Lamm_gx(G2),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.atLeast1_atMost_eq
tff(fact_6215_prod_OatLeast1__atMost__eq,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_gm(G2)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.atLeast1_atMost_eq
tff(fact_6216_prod__mono,axiom,
    ! [A4: set_complex,F2: fun_complex_real,G2: fun_complex_real] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F2,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F2,I3)),aa_complex_real(G2,I3))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups766887009212190081x_real(F2,A4)),groups766887009212190081x_real(G2,A4))) ) ).

% prod_mono
tff(fact_6217_prod__mono,axiom,
    ! [A4: set_real,F2: fun_real_real,G2: fun_real_real] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F2,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F2,I3)),aa_real_real(G2,I3))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F2,A4)),groups1681761925125756287l_real(G2,A4))) ) ).

% prod_mono
tff(fact_6218_prod__mono,axiom,
    ! [A4: set_nat,F2: fun_nat_real,G2: fun_nat_real] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F2,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F2,I3)),aa_nat_real(G2,I3))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F2),A4)),aa_set_nat_real(groups129246275422532515t_real(G2),A4))) ) ).

% prod_mono
tff(fact_6219_prod__mono,axiom,
    ! [A4: set_int,F2: fun_int_real,G2: fun_int_real] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F2,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F2,I3)),aa_int_real(G2,I3))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F2,A4)),groups2316167850115554303t_real(G2,A4))) ) ).

% prod_mono
tff(fact_6220_prod__mono,axiom,
    ! [A4: set_complex,F2: fun_complex_rat,G2: fun_complex_rat] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A4))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F2,I3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F2,I3)),aa_complex_rat(G2,I3))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups225925009352817453ex_rat(F2,A4)),groups225925009352817453ex_rat(G2,A4))) ) ).

% prod_mono
tff(fact_6221_prod__mono,axiom,
    ! [A4: set_real,F2: fun_real_rat,G2: fun_real_rat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F2,I3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F2,I3)),aa_real_rat(G2,I3))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F2,A4)),groups4061424788464935467al_rat(G2,A4))) ) ).

% prod_mono
tff(fact_6222_prod__mono,axiom,
    ! [A4: set_nat,F2: fun_nat_rat,G2: fun_nat_rat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A4))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F2,I3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,I3)),aa_nat_rat(G2,I3))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups73079841787564623at_rat(F2,A4)),groups73079841787564623at_rat(G2,A4))) ) ).

% prod_mono
tff(fact_6223_prod__mono,axiom,
    ! [A4: set_int,F2: fun_int_rat,G2: fun_int_rat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F2,I3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F2,I3)),aa_int_rat(G2,I3))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1072433553688619179nt_rat(F2,A4)),groups1072433553688619179nt_rat(G2,A4))) ) ).

% prod_mono
tff(fact_6224_prod__mono,axiom,
    ! [A4: set_complex,F2: fun_complex_nat,G2: fun_complex_nat] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A4))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F2,I3)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F2,I3)),aa_complex_nat(G2,I3))) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups861055069439313189ex_nat(F2,A4)),groups861055069439313189ex_nat(G2,A4))) ) ).

% prod_mono
tff(fact_6225_prod__mono,axiom,
    ! [A4: set_real,F2: fun_real_nat,G2: fun_real_nat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F2,I3)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F2,I3)),aa_real_nat(G2,I3))) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4696554848551431203al_nat(F2,A4)),groups4696554848551431203al_nat(G2,A4))) ) ).

% prod_mono
tff(fact_6226_prod__nonneg,axiom,
    ! [A4: set_nat,F2: fun_nat_int] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F2,X3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups705719431365010083at_int(F2,A4))) ) ).

% prod_nonneg
tff(fact_6227_prod__nonneg,axiom,
    ! [A4: set_int,F2: fun_int_int] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(F2,X3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F2,A4))) ) ).

% prod_nonneg
tff(fact_6228_prod__nonneg,axiom,
    ! [A4: set_nat,F2: fun_nat_nat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F2,X3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F2),A4))) ) ).

% prod_nonneg
tff(fact_6229_prod__pos,axiom,
    ! [A4: set_nat,F2: fun_nat_int] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F2,X3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),groups705719431365010083at_int(F2,A4))) ) ).

% prod_pos
tff(fact_6230_prod__pos,axiom,
    ! [A4: set_int,F2: fun_int_int] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(F2,X3))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F2,A4))) ) ).

% prod_pos
tff(fact_6231_prod__pos,axiom,
    ! [A4: set_nat,F2: fun_nat_nat] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F2,X3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F2),A4))) ) ).

% prod_pos
tff(fact_6232_prod__ge__1,axiom,
    ! [A4: set_complex,F2: fun_co8153928105989009288nteger] :
      ( ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),A4))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_com4217951221296830281nteger(F2,X3))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),groups8682486955453173170nteger(F2,A4))) ) ).

% prod_ge_1
tff(fact_6233_prod__ge__1,axiom,
    ! [A4: set_real,F2: fun_re6978948838062957830nteger] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A4))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_real_Code_integer(F2,X3))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),groups6225526099057966256nteger(F2,A4))) ) ).

% prod_ge_1
tff(fact_6234_prod__ge__1,axiom,
    ! [A4: set_nat,F2: fun_nat_Code_integer] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A4))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(F2,X3))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),groups3455450783089532116nteger(F2,A4))) ) ).

% prod_ge_1
tff(fact_6235_prod__ge__1,axiom,
    ! [A4: set_int,F2: fun_int_Code_integer] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A4))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(F2,X3))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),groups3827104343326376752nteger(F2,A4))) ) ).

% prod_ge_1
tff(fact_6236_prod__ge__1,axiom,
    ! [A4: set_complex,F2: fun_complex_real] :
      ( ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F2,X3))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups766887009212190081x_real(F2,A4))) ) ).

% prod_ge_1
tff(fact_6237_prod__ge__1,axiom,
    ! [A4: set_real,F2: fun_real_real] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F2,X3))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F2,A4))) ) ).

% prod_ge_1
tff(fact_6238_prod__ge__1,axiom,
    ! [A4: set_nat,F2: fun_nat_real] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F2,X3))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F2),A4))) ) ).

% prod_ge_1
tff(fact_6239_prod__ge__1,axiom,
    ! [A4: set_int,F2: fun_int_real] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F2,X3))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F2,A4))) ) ).

% prod_ge_1
tff(fact_6240_prod__ge__1,axiom,
    ! [A4: set_complex,F2: fun_complex_rat] :
      ( ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_complex_rat(F2,X3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups225925009352817453ex_rat(F2,A4))) ) ).

% prod_ge_1
tff(fact_6241_prod__ge__1,axiom,
    ! [A4: set_real,F2: fun_real_rat] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F2,X3))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups4061424788464935467al_rat(F2,A4))) ) ).

% prod_ge_1
tff(fact_6242_prod__zero,axiom,
    ! [A4: set_nat,F2: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A4))
            & ( aa_nat_complex(F2,X5) = zero_zero_complex ) )
       => ( aa_set_nat_complex(groups6464643781859351333omplex(F2),A4) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_6243_prod__zero,axiom,
    ! [A4: set_int,F2: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A4))
            & ( aa_int_complex(F2,X5) = zero_zero_complex ) )
       => ( groups7440179247065528705omplex(F2,A4) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_6244_prod__zero,axiom,
    ! [A4: set_complex,F2: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A4))
            & ( aa_complex_complex(F2,X5) = zero_zero_complex ) )
       => ( groups3708469109370488835omplex(F2,A4) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_6245_prod__zero,axiom,
    ! [A4: set_nat,F2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A4))
            & ( aa_nat_real(F2,X5) = zero_zero_real ) )
       => ( aa_set_nat_real(groups129246275422532515t_real(F2),A4) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_6246_prod__zero,axiom,
    ! [A4: set_int,F2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A4))
            & ( aa_int_real(F2,X5) = zero_zero_real ) )
       => ( groups2316167850115554303t_real(F2,A4) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_6247_prod__zero,axiom,
    ! [A4: set_complex,F2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A4))
            & ( aa_complex_real(F2,X5) = zero_zero_real ) )
       => ( groups766887009212190081x_real(F2,A4) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_6248_prod__zero,axiom,
    ! [A4: set_nat,F2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A4))
            & ( aa_nat_rat(F2,X5) = zero_zero_rat ) )
       => ( groups73079841787564623at_rat(F2,A4) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_6249_prod__zero,axiom,
    ! [A4: set_int,F2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A4))
            & ( aa_int_rat(F2,X5) = zero_zero_rat ) )
       => ( groups1072433553688619179nt_rat(F2,A4) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_6250_prod__zero,axiom,
    ! [A4: set_complex,F2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ? [X5: complex] :
            ( pp(aa_set_complex_bool(member_complex(X5),A4))
            & ( aa_complex_rat(F2,X5) = zero_zero_rat ) )
       => ( groups225925009352817453ex_rat(F2,A4) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_6251_prod__zero,axiom,
    ! [A4: set_int,F2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A4))
            & ( aa_int_nat(F2,X5) = zero_zero_nat ) )
       => ( groups1707563613775114915nt_nat(F2,A4) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_6252_prod__atLeastAtMost__code,axiom,
    ! [F2: fun_nat_Code_integer,A: nat,B: nat] : groups3455450783089532116nteger(F2,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_fo1084959871951514735nteger(aTP_Lamm_lm(F2),A,B,one_one_Code_integer) ).

% prod_atLeastAtMost_code
tff(fact_6253_prod__atLeastAtMost__code,axiom,
    ! [F2: fun_nat_complex,A: nat,B: nat] : aa_set_nat_complex(groups6464643781859351333omplex(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_ln(F2),A,B),one_one_complex) ).

% prod_atLeastAtMost_code
tff(fact_6254_prod__atLeastAtMost__code,axiom,
    ! [F2: fun_nat_real,A: nat,B: nat] : aa_set_nat_real(groups129246275422532515t_real(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_lo(F2),A),B),one_one_real) ).

% prod_atLeastAtMost_code
tff(fact_6255_prod__atLeastAtMost__code,axiom,
    ! [F2: fun_nat_rat,A: nat,B: nat] : groups73079841787564623at_rat(F2,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_lp(F2),A,B),one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_6256_prod__atLeastAtMost__code,axiom,
    ! [F2: fun_nat_int,A: nat,B: nat] : groups705719431365010083at_int(F2,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_lq(F2),A,B),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_6257_prod__atLeastAtMost__code,axiom,
    ! [F2: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups708209901874060359at_nat(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_lr(F2),A),B),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_6258_Iio__eq__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
    <=> ( N = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
tff(fact_6259_lessThan__strict__subset__iff,axiom,
    ! [M2: real,N: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(set_or5984915006950818249n_real(M2)),set_or5984915006950818249n_real(N)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,M2),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_6260_lessThan__strict__subset__iff,axiom,
    ! [M2: rat,N: rat] :
      ( ord_less_set_rat(set_ord_lessThan_rat(M2),set_ord_lessThan_rat(N))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,M2),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_6261_lessThan__strict__subset__iff,axiom,
    ! [M2: num,N: num] :
      ( ord_less_set_num(set_ord_lessThan_num(M2),set_ord_lessThan_num(N))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_6262_lessThan__strict__subset__iff,axiom,
    ! [M2: int,N: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_ord_lessThan_int(M2)),set_ord_lessThan_int(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M2),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_6263_lessThan__strict__subset__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,M2)),aa_nat_set_nat(set_ord_lessThan_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_6264_lessThan__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K2)) = insert_nat(K2,aa_nat_set_nat(set_ord_lessThan_nat,K2)) ).

% lessThan_Suc
tff(fact_6265_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
    <=> ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_6266_dbl__def,axiom,
    ! [X: real] : aa_real_real(neg_numeral_dbl_real,X) = aa_real_real(plus_plus_real(X),X) ).

% dbl_def
tff(fact_6267_dbl__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_numeral_dbl_rat,X) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X) ).

% dbl_def
tff(fact_6268_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_6269_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G2: fun_nat_int,M2: nat,N: nat] : groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),aa_nat_nat(suc,N))) = groups705719431365010083at_int(aTP_Lamm_gx(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_6270_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G2: fun_nat_nat,M2: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_gm(G2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_6271_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G2: fun_nat_int,M2: nat,K2: nat,N: nat] : groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K2))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ls(G2),K2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_6272_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G2: fun_nat_nat,M2: nat,K2: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K2))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_go(G2),K2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_6273_prod__le__1,axiom,
    ! [A4: set_complex,F2: fun_co8153928105989009288nteger] :
      ( ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),A4))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(F2,X3)))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_com4217951221296830281nteger(F2,X3)),one_one_Code_integer)) ) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups8682486955453173170nteger(F2,A4)),one_one_Code_integer)) ) ).

% prod_le_1
tff(fact_6274_prod__le__1,axiom,
    ! [A4: set_real,F2: fun_re6978948838062957830nteger] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A4))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(F2,X3)))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_real_Code_integer(F2,X3)),one_one_Code_integer)) ) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups6225526099057966256nteger(F2,A4)),one_one_Code_integer)) ) ).

% prod_le_1
tff(fact_6275_prod__le__1,axiom,
    ! [A4: set_nat,F2: fun_nat_Code_integer] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A4))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F2,X3)))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(F2,X3)),one_one_Code_integer)) ) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups3455450783089532116nteger(F2,A4)),one_one_Code_integer)) ) ).

% prod_le_1
tff(fact_6276_prod__le__1,axiom,
    ! [A4: set_int,F2: fun_int_Code_integer] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A4))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F2,X3)))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(F2,X3)),one_one_Code_integer)) ) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups3827104343326376752nteger(F2,A4)),one_one_Code_integer)) ) ).

% prod_le_1
tff(fact_6277_prod__le__1,axiom,
    ! [A4: set_complex,F2: fun_complex_real] :
      ( ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F2,X3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F2,X3)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups766887009212190081x_real(F2,A4)),one_one_real)) ) ).

% prod_le_1
tff(fact_6278_prod__le__1,axiom,
    ! [A4: set_real,F2: fun_real_real] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F2,X3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F2,X3)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F2,A4)),one_one_real)) ) ).

% prod_le_1
tff(fact_6279_prod__le__1,axiom,
    ! [A4: set_nat,F2: fun_nat_real] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F2,X3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F2,X3)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F2),A4)),one_one_real)) ) ).

% prod_le_1
tff(fact_6280_prod__le__1,axiom,
    ! [A4: set_int,F2: fun_int_real] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F2,X3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F2,X3)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F2,A4)),one_one_real)) ) ).

% prod_le_1
tff(fact_6281_prod__le__1,axiom,
    ! [A4: set_complex,F2: fun_complex_rat] :
      ( ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),A4))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F2,X3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F2,X3)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups225925009352817453ex_rat(F2,A4)),one_one_rat)) ) ).

% prod_le_1
tff(fact_6282_prod__le__1,axiom,
    ! [A4: set_real,F2: fun_real_rat] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),A4))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F2,X3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F2,X3)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F2,A4)),one_one_rat)) ) ).

% prod_le_1
tff(fact_6283_prod_Orelated,axiom,
    ! [R2: fun_Co8310303586393106575r_bool,S3: set_nat,H: fun_nat_Code_integer,G2: fun_nat_Code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,one_one_Code_integer),one_one_Code_integer))
     => ( ! [X15: code_integer,Y15: code_integer,X23: code_integer,Y23: code_integer] :
            ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,X15),X23))
              & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,Y15),Y23)) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X15),Y15)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S3))
         => ( ! [X3: nat] :
                ( pp(aa_set_nat_bool(member_nat(X3),S3))
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,aa_nat_Code_integer(H,X3)),aa_nat_Code_integer(G2,X3))) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,groups3455450783089532116nteger(H,S3)),groups3455450783089532116nteger(G2,S3))) ) ) ) ) ).

% prod.related
tff(fact_6284_prod_Orelated,axiom,
    ! [R2: fun_Co8310303586393106575r_bool,S3: set_int,H: fun_int_Code_integer,G2: fun_int_Code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,one_one_Code_integer),one_one_Code_integer))
     => ( ! [X15: code_integer,Y15: code_integer,X23: code_integer,Y23: code_integer] :
            ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,X15),X23))
              & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,Y15),Y23)) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X15),Y15)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S3))
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,aa_int_Code_integer(H,X3)),aa_int_Code_integer(G2,X3))) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,groups3827104343326376752nteger(H,S3)),groups3827104343326376752nteger(G2,S3))) ) ) ) ) ).

% prod.related
tff(fact_6285_prod_Orelated,axiom,
    ! [R2: fun_Co8310303586393106575r_bool,S3: set_complex,H: fun_co8153928105989009288nteger,G2: fun_co8153928105989009288nteger] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,one_one_Code_integer),one_one_Code_integer))
     => ( ! [X15: code_integer,Y15: code_integer,X23: code_integer,Y23: code_integer] :
            ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,X15),X23))
              & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,Y15),Y23)) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X15),Y15)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X3: complex] :
                ( pp(aa_set_complex_bool(member_complex(X3),S3))
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,aa_com4217951221296830281nteger(H,X3)),aa_com4217951221296830281nteger(G2,X3))) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R2,groups8682486955453173170nteger(H,S3)),groups8682486955453173170nteger(G2,S3))) ) ) ) ) ).

% prod.related
tff(fact_6286_prod_Orelated,axiom,
    ! [R2: fun_co1378794544173401837x_bool,S3: set_nat,H: fun_nat_complex,G2: fun_nat_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,one_one_complex),one_one_complex))
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X15),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,Y15),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S3))
         => ( ! [X3: nat] :
                ( pp(aa_set_nat_bool(member_nat(X3),S3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_nat_complex(H,X3)),aa_nat_complex(G2,X3))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_set_nat_complex(groups6464643781859351333omplex(H),S3)),aa_set_nat_complex(groups6464643781859351333omplex(G2),S3))) ) ) ) ) ).

% prod.related
tff(fact_6287_prod_Orelated,axiom,
    ! [R2: fun_co1378794544173401837x_bool,S3: set_int,H: fun_int_complex,G2: fun_int_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,one_one_complex),one_one_complex))
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X15),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,Y15),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_int_complex(H,X3)),aa_int_complex(G2,X3))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,groups7440179247065528705omplex(H,S3)),groups7440179247065528705omplex(G2,S3))) ) ) ) ) ).

% prod.related
tff(fact_6288_prod_Orelated,axiom,
    ! [R2: fun_co1378794544173401837x_bool,S3: set_complex,H: fun_complex_complex,G2: fun_complex_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,one_one_complex),one_one_complex))
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X15),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,Y15),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X3: complex] :
                ( pp(aa_set_complex_bool(member_complex(X3),S3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_complex_complex(H,X3)),aa_complex_complex(G2,X3))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,groups3708469109370488835omplex(H,S3)),groups3708469109370488835omplex(G2,S3))) ) ) ) ) ).

% prod.related
tff(fact_6289_prod_Orelated,axiom,
    ! [R2: fun_re1255401033344146673l_bool,S3: set_nat,H: fun_nat_real,G2: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,one_one_real),one_one_real))
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,Y15),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S3))
         => ( ! [X3: nat] :
                ( pp(aa_set_nat_bool(member_nat(X3),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_nat_real(H,X3)),aa_nat_real(G2,X3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_set_nat_real(groups129246275422532515t_real(H),S3)),aa_set_nat_real(groups129246275422532515t_real(G2),S3))) ) ) ) ) ).

% prod.related
tff(fact_6290_prod_Orelated,axiom,
    ! [R2: fun_re1255401033344146673l_bool,S3: set_int,H: fun_int_real,G2: fun_int_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,one_one_real),one_one_real))
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,Y15),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X3: int] :
                ( pp(aa_set_int_bool(member_int(X3),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_int_real(H,X3)),aa_int_real(G2,X3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,groups2316167850115554303t_real(H,S3)),groups2316167850115554303t_real(G2,S3))) ) ) ) ) ).

% prod.related
tff(fact_6291_prod_Orelated,axiom,
    ! [R2: fun_re1255401033344146673l_bool,S3: set_complex,H: fun_complex_real,G2: fun_complex_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,one_one_real),one_one_real))
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,Y15),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X3: complex] :
                ( pp(aa_set_complex_bool(member_complex(X3),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_complex_real(H,X3)),aa_complex_real(G2,X3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,groups766887009212190081x_real(H,S3)),groups766887009212190081x_real(G2,S3))) ) ) ) ) ).

% prod.related
tff(fact_6292_prod_Orelated,axiom,
    ! [R2: fun_rat_fun_rat_bool,S3: set_nat,H: fun_nat_rat,G2: fun_nat_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,one_one_rat),one_one_rat))
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,Y15),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X15),Y15)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S3))
         => ( ! [X3: nat] :
                ( pp(aa_set_nat_bool(member_nat(X3),S3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_nat_rat(H,X3)),aa_nat_rat(G2,X3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,groups73079841787564623at_rat(H,S3)),groups73079841787564623at_rat(G2,S3))) ) ) ) ) ).

% prod.related
tff(fact_6293_prod_Oinsert__if,axiom,
    ! [A4: set_real,X: real,G2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( pp(aa_set_real_bool(member_real(X),A4))
         => ( groups1681761925125756287l_real(G2,insert_real(X,A4)) = groups1681761925125756287l_real(G2,A4) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A4))
         => ( groups1681761925125756287l_real(G2,insert_real(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G2,X)),groups1681761925125756287l_real(G2,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_6294_prod_Oinsert__if,axiom,
    ! [A4: set_nat,X: nat,G2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( pp(aa_set_nat_bool(member_nat(X),A4))
         => ( aa_set_nat_real(groups129246275422532515t_real(G2),insert_nat(X,A4)) = aa_set_nat_real(groups129246275422532515t_real(G2),A4) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
         => ( aa_set_nat_real(groups129246275422532515t_real(G2),insert_nat(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G2,X)),aa_set_nat_real(groups129246275422532515t_real(G2),A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_6295_prod_Oinsert__if,axiom,
    ! [A4: set_int,X: int,G2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( pp(aa_set_int_bool(member_int(X),A4))
         => ( groups2316167850115554303t_real(G2,insert_int(X,A4)) = groups2316167850115554303t_real(G2,A4) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A4))
         => ( groups2316167850115554303t_real(G2,insert_int(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G2,X)),groups2316167850115554303t_real(G2,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_6296_prod_Oinsert__if,axiom,
    ! [A4: set_complex,X: complex,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A4))
         => ( groups766887009212190081x_real(G2,insert_complex(X,A4)) = groups766887009212190081x_real(G2,A4) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A4))
         => ( groups766887009212190081x_real(G2,insert_complex(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G2,X)),groups766887009212190081x_real(G2,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_6297_prod_Oinsert__if,axiom,
    ! [A4: set_real,X: real,G2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( pp(aa_set_real_bool(member_real(X),A4))
         => ( groups4061424788464935467al_rat(G2,insert_real(X,A4)) = groups4061424788464935467al_rat(G2,A4) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A4))
         => ( groups4061424788464935467al_rat(G2,insert_real(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(G2,X)),groups4061424788464935467al_rat(G2,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_6298_prod_Oinsert__if,axiom,
    ! [A4: set_nat,X: nat,G2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( pp(aa_set_nat_bool(member_nat(X),A4))
         => ( groups73079841787564623at_rat(G2,insert_nat(X,A4)) = groups73079841787564623at_rat(G2,A4) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A4))
         => ( groups73079841787564623at_rat(G2,insert_nat(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G2,X)),groups73079841787564623at_rat(G2,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_6299_prod_Oinsert__if,axiom,
    ! [A4: set_int,X: int,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( pp(aa_set_int_bool(member_int(X),A4))
         => ( groups1072433553688619179nt_rat(G2,insert_int(X,A4)) = groups1072433553688619179nt_rat(G2,A4) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A4))
         => ( groups1072433553688619179nt_rat(G2,insert_int(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(G2,X)),groups1072433553688619179nt_rat(G2,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_6300_prod_Oinsert__if,axiom,
    ! [A4: set_complex,X: complex,G2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( pp(aa_set_complex_bool(member_complex(X),A4))
         => ( groups225925009352817453ex_rat(G2,insert_complex(X,A4)) = groups225925009352817453ex_rat(G2,A4) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(X),A4))
         => ( groups225925009352817453ex_rat(G2,insert_complex(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_complex_rat(G2,X)),groups225925009352817453ex_rat(G2,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_6301_prod_Oinsert__if,axiom,
    ! [A4: set_real,X: real,G2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( pp(aa_set_real_bool(member_real(X),A4))
         => ( groups4696554848551431203al_nat(G2,insert_real(X,A4)) = groups4696554848551431203al_nat(G2,A4) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A4))
         => ( groups4696554848551431203al_nat(G2,insert_real(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_real_nat(G2,X)),groups4696554848551431203al_nat(G2,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_6302_prod_Oinsert__if,axiom,
    ! [A4: set_int,X: int,G2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( pp(aa_set_int_bool(member_int(X),A4))
         => ( groups1707563613775114915nt_nat(G2,insert_int(X,A4)) = groups1707563613775114915nt_nat(G2,A4) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A4))
         => ( groups1707563613775114915nt_nat(G2,insert_int(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(G2,X)),groups1707563613775114915nt_nat(G2,A4)) ) ) ) ) ).

% prod.insert_if
tff(fact_6303_ivl__disj__int__one_I4_J,axiom,
    ! [L: nat,U: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,L)),aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = bot_bot_set_nat ).

% ivl_disj_int_one(4)
tff(fact_6304_ivl__disj__int__one_I4_J,axiom,
    ! [L: int,U: int] : inf_inf_set_int(set_ord_lessThan_int(L),set_or1266510415728281911st_int(L,U)) = bot_bot_set_int ).

% ivl_disj_int_one(4)
tff(fact_6305_ivl__disj__int__one_I4_J,axiom,
    ! [L: real,U: real] : inf_inf_set_real(set_or5984915006950818249n_real(L),set_or1222579329274155063t_real(L,U)) = bot_bot_set_real ).

% ivl_disj_int_one(4)
tff(fact_6306_prod_OatLeastAtMost__rev,axiom,
    ! [G2: fun_nat_int,N: nat,M2: nat] : groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(N),M2)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_lt(G2),N),M2),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M2)) ).

% prod.atLeastAtMost_rev
tff(fact_6307_prod_OatLeastAtMost__rev,axiom,
    ! [G2: fun_nat_nat,N: nat,M2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_gu(G2),N),M2)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M2)) ).

% prod.atLeastAtMost_rev
tff(fact_6308_sum_Onat__diff__reindex,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ld(G2),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nat_diff_reindex
tff(fact_6309_sum_Onat__diff__reindex,axiom,
    ! [G2: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_lu(G2),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nat_diff_reindex
tff(fact_6310_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P2: fun_nat_nat,N: nat] :
      ( ! [X3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Q,X3)),aa_nat_nat(P2,X3)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ip(Q),P2)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ) ).

% sum_diff_distrib
tff(fact_6311_less__1__prod2,axiom,
    ! [I5: set_real,I2: real,F2: fun_re6978948838062957830nteger] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( pp(aa_set_real_bool(member_real(I2),I5))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_real_Code_integer(F2,I2)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I5))
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_real_Code_integer(F2,I3))) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),groups6225526099057966256nteger(F2,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_6312_less__1__prod2,axiom,
    ! [I5: set_nat,I2: nat,F2: fun_nat_Code_integer] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( pp(aa_set_nat_bool(member_nat(I2),I5))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(F2,I2)))
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(F2,I3))) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),groups3455450783089532116nteger(F2,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_6313_less__1__prod2,axiom,
    ! [I5: set_int,I2: int,F2: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( pp(aa_set_int_bool(member_int(I2),I5))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(F2,I2)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I5))
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(F2,I3))) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),groups3827104343326376752nteger(F2,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_6314_less__1__prod2,axiom,
    ! [I5: set_complex,I2: complex,F2: fun_co8153928105989009288nteger] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( pp(aa_set_complex_bool(member_complex(I2),I5))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_com4217951221296830281nteger(F2,I2)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I5))
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_com4217951221296830281nteger(F2,I3))) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),groups8682486955453173170nteger(F2,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_6315_less__1__prod2,axiom,
    ! [I5: set_real,I2: real,F2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( pp(aa_set_real_bool(member_real(I2),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(F2,I2)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F2,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups1681761925125756287l_real(F2,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_6316_less__1__prod2,axiom,
    ! [I5: set_nat,I2: nat,F2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( 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(F2,I2)))
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F2,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F2),I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_6317_less__1__prod2,axiom,
    ! [I5: set_int,I2: int,F2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( 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(F2,I2)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F2,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2316167850115554303t_real(F2,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_6318_less__1__prod2,axiom,
    ! [I5: set_complex,I2: complex,F2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( 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(F2,I2)))
         => ( ! [I3: complex] :
                ( pp(aa_set_complex_bool(member_complex(I3),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F2,I3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups766887009212190081x_real(F2,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_6319_less__1__prod2,axiom,
    ! [I5: set_real,I2: real,F2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( pp(aa_set_real_bool(member_real(I2),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_real_rat(F2,I2)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F2,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F2,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_6320_less__1__prod2,axiom,
    ! [I5: set_nat,I2: nat,F2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( 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(F2,I2)))
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(F2,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F2,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_6321_prod_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G2: fun_int_real] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B5),A4))
     => ( pp(aa_set_int_bool(finite_finite_int,A4))
       => ( groups2316167850115554303t_real(G2,A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups2316167850115554303t_real(G2,minus_minus_set_int(A4,B5))),groups2316167850115554303t_real(G2,B5)) ) ) ) ).

% prod.subset_diff
tff(fact_6322_prod_Osubset__diff,axiom,
    ! [B5: set_complex,A4: set_complex,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B5),A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
       => ( groups766887009212190081x_real(G2,A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(G2,minus_811609699411566653omplex(A4,B5))),groups766887009212190081x_real(G2,B5)) ) ) ) ).

% prod.subset_diff
tff(fact_6323_prod_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B5),A4))
     => ( pp(aa_set_int_bool(finite_finite_int,A4))
       => ( groups1072433553688619179nt_rat(G2,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G2,minus_minus_set_int(A4,B5))),groups1072433553688619179nt_rat(G2,B5)) ) ) ) ).

% prod.subset_diff
tff(fact_6324_prod_Osubset__diff,axiom,
    ! [B5: set_complex,A4: set_complex,G2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B5),A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
       => ( groups225925009352817453ex_rat(G2,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups225925009352817453ex_rat(G2,minus_811609699411566653omplex(A4,B5))),groups225925009352817453ex_rat(G2,B5)) ) ) ) ).

% prod.subset_diff
tff(fact_6325_prod_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G2: fun_int_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B5),A4))
     => ( pp(aa_set_int_bool(finite_finite_int,A4))
       => ( groups1707563613775114915nt_nat(G2,A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G2,minus_minus_set_int(A4,B5))),groups1707563613775114915nt_nat(G2,B5)) ) ) ) ).

% prod.subset_diff
tff(fact_6326_prod_Osubset__diff,axiom,
    ! [B5: set_complex,A4: set_complex,G2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B5),A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
       => ( groups861055069439313189ex_nat(G2,A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G2,minus_811609699411566653omplex(A4,B5))),groups861055069439313189ex_nat(G2,B5)) ) ) ) ).

% prod.subset_diff
tff(fact_6327_prod_Osubset__diff,axiom,
    ! [B5: set_complex,A4: set_complex,G2: fun_complex_int] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B5),A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
       => ( groups858564598930262913ex_int(G2,A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups858564598930262913ex_int(G2,minus_811609699411566653omplex(A4,B5))),groups858564598930262913ex_int(G2,B5)) ) ) ) ).

% prod.subset_diff
tff(fact_6328_prod_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G2: fun_nat_real] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B5),A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A4))
       => ( aa_set_nat_real(groups129246275422532515t_real(G2),A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B5))),aa_set_nat_real(groups129246275422532515t_real(G2),B5)) ) ) ) ).

% prod.subset_diff
tff(fact_6329_prod_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B5),A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A4))
       => ( groups73079841787564623at_rat(G2,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B5))),groups73079841787564623at_rat(G2,B5)) ) ) ) ).

% prod.subset_diff
tff(fact_6330_prod_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G2: fun_nat_int] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B5),A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A4))
       => ( groups705719431365010083at_int(G2,A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B5))),groups705719431365010083at_int(G2,B5)) ) ) ) ).

% prod.subset_diff
tff(fact_6331_prod_Ounion__inter,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups2316167850115554303t_real(G2,sup_sup_set_int(A4,B5))),groups2316167850115554303t_real(G2,inf_inf_set_int(A4,B5))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups2316167850115554303t_real(G2,A4)),groups2316167850115554303t_real(G2,B5)) ) ) ) ).

% prod.union_inter
tff(fact_6332_prod_Ounion__inter,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(G2,sup_sup_set_complex(A4,B5))),groups766887009212190081x_real(G2,inf_inf_set_complex(A4,B5))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(G2,A4)),groups766887009212190081x_real(G2,B5)) ) ) ) ).

% prod.union_inter
tff(fact_6333_prod_Ounion__inter,axiom,
    ! [A4: set_nat,B5: set_nat,G2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5))),aa_set_nat_real(groups129246275422532515t_real(G2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G2),A4)),aa_set_nat_real(groups129246275422532515t_real(G2),B5)) ) ) ) ).

% prod.union_inter
tff(fact_6334_prod_Ounion__inter,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G2,sup_sup_set_int(A4,B5))),groups1072433553688619179nt_rat(G2,inf_inf_set_int(A4,B5))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G2,A4)),groups1072433553688619179nt_rat(G2,B5)) ) ) ) ).

% prod.union_inter
tff(fact_6335_prod_Ounion__inter,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups225925009352817453ex_rat(G2,sup_sup_set_complex(A4,B5))),groups225925009352817453ex_rat(G2,inf_inf_set_complex(A4,B5))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups225925009352817453ex_rat(G2,A4)),groups225925009352817453ex_rat(G2,B5)) ) ) ) ).

% prod.union_inter
tff(fact_6336_prod_Ounion__inter,axiom,
    ! [A4: set_nat,B5: set_nat,G2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5))),groups73079841787564623at_rat(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G2,A4)),groups73079841787564623at_rat(G2,B5)) ) ) ) ).

% prod.union_inter
tff(fact_6337_prod_Ounion__inter,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G2,sup_sup_set_int(A4,B5))),groups1707563613775114915nt_nat(G2,inf_inf_set_int(A4,B5))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G2,A4)),groups1707563613775114915nt_nat(G2,B5)) ) ) ) ).

% prod.union_inter
tff(fact_6338_prod_Ounion__inter,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G2,sup_sup_set_complex(A4,B5))),groups861055069439313189ex_nat(G2,inf_inf_set_complex(A4,B5))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G2,A4)),groups861055069439313189ex_nat(G2,B5)) ) ) ) ).

% prod.union_inter
tff(fact_6339_prod_Ounion__inter,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups858564598930262913ex_int(G2,sup_sup_set_complex(A4,B5))),groups858564598930262913ex_int(G2,inf_inf_set_complex(A4,B5))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups858564598930262913ex_int(G2,A4)),groups858564598930262913ex_int(G2,B5)) ) ) ) ).

% prod.union_inter
tff(fact_6340_prod_Ounion__inter,axiom,
    ! [A4: set_nat,B5: set_nat,G2: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5))),groups705719431365010083at_int(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G2,A4)),groups705719431365010083at_int(G2,B5)) ) ) ) ).

% prod.union_inter
tff(fact_6341_prod_OInt__Diff,axiom,
    ! [A4: set_int,G2: fun_int_real,B5: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups2316167850115554303t_real(G2,A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups2316167850115554303t_real(G2,inf_inf_set_int(A4,B5))),groups2316167850115554303t_real(G2,minus_minus_set_int(A4,B5))) ) ) ).

% prod.Int_Diff
tff(fact_6342_prod_OInt__Diff,axiom,
    ! [A4: set_complex,G2: fun_complex_real,B5: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups766887009212190081x_real(G2,A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(G2,inf_inf_set_complex(A4,B5))),groups766887009212190081x_real(G2,minus_811609699411566653omplex(A4,B5))) ) ) ).

% prod.Int_Diff
tff(fact_6343_prod_OInt__Diff,axiom,
    ! [A4: set_nat,G2: fun_nat_real,B5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( aa_set_nat_real(groups129246275422532515t_real(G2),A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5))),aa_set_nat_real(groups129246275422532515t_real(G2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B5))) ) ) ).

% prod.Int_Diff
tff(fact_6344_prod_OInt__Diff,axiom,
    ! [A4: set_int,G2: fun_int_rat,B5: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups1072433553688619179nt_rat(G2,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G2,inf_inf_set_int(A4,B5))),groups1072433553688619179nt_rat(G2,minus_minus_set_int(A4,B5))) ) ) ).

% prod.Int_Diff
tff(fact_6345_prod_OInt__Diff,axiom,
    ! [A4: set_complex,G2: fun_complex_rat,B5: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups225925009352817453ex_rat(G2,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups225925009352817453ex_rat(G2,inf_inf_set_complex(A4,B5))),groups225925009352817453ex_rat(G2,minus_811609699411566653omplex(A4,B5))) ) ) ).

% prod.Int_Diff
tff(fact_6346_prod_OInt__Diff,axiom,
    ! [A4: set_nat,G2: fun_nat_rat,B5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( groups73079841787564623at_rat(G2,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5))),groups73079841787564623at_rat(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B5))) ) ) ).

% prod.Int_Diff
tff(fact_6347_prod_OInt__Diff,axiom,
    ! [A4: set_int,G2: fun_int_nat,B5: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups1707563613775114915nt_nat(G2,A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G2,inf_inf_set_int(A4,B5))),groups1707563613775114915nt_nat(G2,minus_minus_set_int(A4,B5))) ) ) ).

% prod.Int_Diff
tff(fact_6348_prod_OInt__Diff,axiom,
    ! [A4: set_complex,G2: fun_complex_nat,B5: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups861055069439313189ex_nat(G2,A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G2,inf_inf_set_complex(A4,B5))),groups861055069439313189ex_nat(G2,minus_811609699411566653omplex(A4,B5))) ) ) ).

% prod.Int_Diff
tff(fact_6349_prod_OInt__Diff,axiom,
    ! [A4: set_complex,G2: fun_complex_int,B5: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups858564598930262913ex_int(G2,A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups858564598930262913ex_int(G2,inf_inf_set_complex(A4,B5))),groups858564598930262913ex_int(G2,minus_811609699411566653omplex(A4,B5))) ) ) ).

% prod.Int_Diff
tff(fact_6350_prod_OInt__Diff,axiom,
    ! [A4: set_nat,G2: fun_nat_int,B5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( groups705719431365010083at_int(G2,A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5))),groups705719431365010083at_int(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B5))) ) ) ).

% prod.Int_Diff
tff(fact_6351_prod_OatLeast0__atMost__Suc,axiom,
    ! [G2: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_real(G2,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6352_prod_OatLeast0__atMost__Suc,axiom,
    ! [G2: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_rat(G2,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6353_prod_OatLeast0__atMost__Suc,axiom,
    ! [G2: fun_nat_int,N: nat] : groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_int(G2,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6354_prod_OatLeast0__atMost__Suc,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_nat(G2,aa_nat_nat(suc,N))) ).

% prod.atLeast0_atMost_Suc
tff(fact_6355_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G2,M2)),aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6356_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( groups73079841787564623at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G2,M2)),groups73079841787564623at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6357_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G2,M2)),groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6358_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G2,M2)),aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),N))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_6359_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G2,aa_nat_nat(suc,N))),aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6360_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( groups73079841787564623at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G2,aa_nat_nat(suc,N))),groups73079841787564623at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6361_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G2,aa_nat_nat(suc,N))),groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6362_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G2,aa_nat_nat(suc,N))),aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_6363_Iio__Int__singleton,axiom,
    ! [X: real,K2: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),K2))
       => ( inf_inf_set_real(set_or5984915006950818249n_real(K2),insert_real(X,bot_bot_set_real)) = insert_real(X,bot_bot_set_real) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),K2))
       => ( inf_inf_set_real(set_or5984915006950818249n_real(K2),insert_real(X,bot_bot_set_real)) = bot_bot_set_real ) ) ) ).

% Iio_Int_singleton
tff(fact_6364_Iio__Int__singleton,axiom,
    ! [X: rat,K2: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),K2))
       => ( inf_inf_set_rat(set_ord_lessThan_rat(K2),insert_rat(X,bot_bot_set_rat)) = insert_rat(X,bot_bot_set_rat) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),K2))
       => ( inf_inf_set_rat(set_ord_lessThan_rat(K2),insert_rat(X,bot_bot_set_rat)) = bot_bot_set_rat ) ) ) ).

% Iio_Int_singleton
tff(fact_6365_Iio__Int__singleton,axiom,
    ! [X: num,K2: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),K2))
       => ( inf_inf_set_num(set_ord_lessThan_num(K2),insert_num(X,bot_bot_set_num)) = insert_num(X,bot_bot_set_num) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),K2))
       => ( inf_inf_set_num(set_ord_lessThan_num(K2),insert_num(X,bot_bot_set_num)) = bot_bot_set_num ) ) ) ).

% Iio_Int_singleton
tff(fact_6366_Iio__Int__singleton,axiom,
    ! [X: int,K2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),K2))
       => ( inf_inf_set_int(set_ord_lessThan_int(K2),insert_int(X,bot_bot_set_int)) = insert_int(X,bot_bot_set_int) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),K2))
       => ( inf_inf_set_int(set_ord_lessThan_int(K2),insert_int(X,bot_bot_set_int)) = bot_bot_set_int ) ) ) ).

% Iio_Int_singleton
tff(fact_6367_Iio__Int__singleton,axiom,
    ! [X: nat,K2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),K2))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,K2)),insert_nat(X,bot_bot_set_nat)) = insert_nat(X,bot_bot_set_nat) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),K2))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,K2)),insert_nat(X,bot_bot_set_nat)) = bot_bot_set_nat ) ) ) ).

% Iio_Int_singleton
tff(fact_6368_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_real(G2,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G2,M2)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_gn(G2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6369_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_rat(G2,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G2,M2)),groups73079841787564623at_rat(aTP_Lamm_gw(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6370_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_int(G2,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G2,M2)),groups705719431365010083at_int(aTP_Lamm_gx(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6371_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_nat(G2,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G2,M2)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_gm(G2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_6372_sum_OlessThan__Suc__shift,axiom,
    ! [G2: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G2,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_gw(G2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6373_sum_OlessThan__Suc__shift,axiom,
    ! [G2: fun_nat_int,N: nat] : groups3539618377306564664at_int(G2,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G2,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_gx(G2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6374_sum_OlessThan__Suc__shift,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G2,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gm(G2)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6375_sum_OlessThan__Suc__shift,axiom,
    ! [G2: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(plus_plus_real(aa_nat_real(G2,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gn(G2)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_6376_sum__lessThan__telescope_H,axiom,
    ! [F2: fun_nat_rat,M2: nat] : groups2906978787729119204at_rat(aTP_Lamm_lv(F2),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F2,zero_zero_nat)),aa_nat_rat(F2,M2)) ).

% sum_lessThan_telescope'
tff(fact_6377_sum__lessThan__telescope_H,axiom,
    ! [F2: fun_nat_int,M2: nat] : groups3539618377306564664at_int(aTP_Lamm_lw(F2),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F2,zero_zero_nat)),aa_nat_int(F2,M2)) ).

% sum_lessThan_telescope'
tff(fact_6378_sum__lessThan__telescope_H,axiom,
    ! [F2: fun_nat_real,M2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lx(F2)),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = aa_real_real(minus_minus_real(aa_nat_real(F2,zero_zero_nat)),aa_nat_real(F2,M2)) ).

% sum_lessThan_telescope'
tff(fact_6379_sum__lessThan__telescope,axiom,
    ! [F2: fun_nat_rat,M2: nat] : groups2906978787729119204at_rat(aTP_Lamm_gy(F2),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F2,M2)),aa_nat_rat(F2,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6380_sum__lessThan__telescope,axiom,
    ! [F2: fun_nat_int,M2: nat] : groups3539618377306564664at_int(aTP_Lamm_gz(F2),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F2,M2)),aa_nat_int(F2,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6381_sum__lessThan__telescope,axiom,
    ! [F2: fun_nat_real,M2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ha(F2)),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = aa_real_real(minus_minus_real(aa_nat_real(F2,M2)),aa_nat_real(F2,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6382_sumr__diff__mult__const2,axiom,
    ! [F2: fun_nat_rat,N: nat,R: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups2906978787729119204at_rat(F2,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),R)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_ly(F2),R),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_6383_sumr__diff__mult__const2,axiom,
    ! [F2: fun_nat_int,N: nat,R: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F2,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),R)) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_lz(F2),R),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_6384_sumr__diff__mult__const2,axiom,
    ! [F2: fun_nat_real,N: nat,R: real] : aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(F2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),R)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ma(F2),R)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_6385_sum_OatLeast1__atMost__eq,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gm(G2)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_6386_sum_OatLeast1__atMost__eq,axiom,
    ! [G2: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gn(G2)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_6387_prod__mono__strict,axiom,
    ! [A4: set_complex,F2: fun_complex_real,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),A4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F2,I3)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F2,I3)),aa_complex_real(G2,I3))) ) )
       => ( ( A4 != bot_bot_set_complex )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups766887009212190081x_real(F2,A4)),groups766887009212190081x_real(G2,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_6388_prod__mono__strict,axiom,
    ! [A4: set_nat,F2: fun_nat_real,G2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),A4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F2,I3)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F2,I3)),aa_nat_real(G2,I3))) ) )
       => ( ( A4 != bot_bot_set_nat )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups129246275422532515t_real(F2),A4)),aa_set_nat_real(groups129246275422532515t_real(G2),A4))) ) ) ) ).

% prod_mono_strict
tff(fact_6389_prod__mono__strict,axiom,
    ! [A4: set_int,F2: fun_int_real,G2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),A4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F2,I3)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F2,I3)),aa_int_real(G2,I3))) ) )
       => ( ( A4 != bot_bot_set_int )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2316167850115554303t_real(F2,A4)),groups2316167850115554303t_real(G2,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_6390_prod__mono__strict,axiom,
    ! [A4: set_real,F2: fun_real_real,G2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),A4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F2,I3)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F2,I3)),aa_real_real(G2,I3))) ) )
       => ( ( A4 != bot_bot_set_real )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups1681761925125756287l_real(F2,A4)),groups1681761925125756287l_real(G2,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_6391_prod__mono__strict,axiom,
    ! [A4: set_complex,F2: fun_complex_rat,G2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),A4))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F2,I3)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F2,I3)),aa_complex_rat(G2,I3))) ) )
       => ( ( A4 != bot_bot_set_complex )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups225925009352817453ex_rat(F2,A4)),groups225925009352817453ex_rat(G2,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_6392_prod__mono__strict,axiom,
    ! [A4: set_nat,F2: fun_nat_rat,G2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),A4))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F2,I3)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F2,I3)),aa_nat_rat(G2,I3))) ) )
       => ( ( A4 != bot_bot_set_nat )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups73079841787564623at_rat(F2,A4)),groups73079841787564623at_rat(G2,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_6393_prod__mono__strict,axiom,
    ! [A4: set_int,F2: fun_int_rat,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),A4))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F2,I3)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F2,I3)),aa_int_rat(G2,I3))) ) )
       => ( ( A4 != bot_bot_set_int )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1072433553688619179nt_rat(F2,A4)),groups1072433553688619179nt_rat(G2,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_6394_prod__mono__strict,axiom,
    ! [A4: set_real,F2: fun_real_rat,G2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),A4))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F2,I3)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F2,I3)),aa_real_rat(G2,I3))) ) )
       => ( ( A4 != bot_bot_set_real )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups4061424788464935467al_rat(F2,A4)),groups4061424788464935467al_rat(G2,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_6395_prod__mono__strict,axiom,
    ! [A4: set_complex,F2: fun_complex_nat,G2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ! [I3: complex] :
            ( pp(aa_set_complex_bool(member_complex(I3),A4))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F2,I3)))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F2,I3)),aa_complex_nat(G2,I3))) ) )
       => ( ( A4 != bot_bot_set_complex )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups861055069439313189ex_nat(F2,A4)),groups861055069439313189ex_nat(G2,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_6396_prod__mono__strict,axiom,
    ! [A4: set_int,F2: fun_int_nat,G2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),A4))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F2,I3)))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F2,I3)),aa_int_nat(G2,I3))) ) )
       => ( ( A4 != bot_bot_set_int )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups1707563613775114915nt_nat(F2,A4)),groups1707563613775114915nt_nat(G2,A4))) ) ) ) ).

% prod_mono_strict
tff(fact_6397_prod_Oremove,axiom,
    ! [A4: set_complex,X: complex,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups766887009212190081x_real(G2,A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G2,X)),groups766887009212190081x_real(G2,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_6398_prod_Oremove,axiom,
    ! [A4: set_int,X: int,G2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(X),A4))
       => ( groups2316167850115554303t_real(G2,A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G2,X)),groups2316167850115554303t_real(G2,minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_6399_prod_Oremove,axiom,
    ! [A4: set_real,X: real,G2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(X),A4))
       => ( groups1681761925125756287l_real(G2,A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G2,X)),groups1681761925125756287l_real(G2,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_6400_prod_Oremove,axiom,
    ! [A4: set_nat,X: nat,G2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(member_nat(X),A4))
       => ( aa_set_nat_real(groups129246275422532515t_real(G2),A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G2,X)),aa_set_nat_real(groups129246275422532515t_real(G2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X,bot_bot_set_nat)))) ) ) ) ).

% prod.remove
tff(fact_6401_prod_Oremove,axiom,
    ! [A4: set_complex,X: complex,G2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups225925009352817453ex_rat(G2,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_complex_rat(G2,X)),groups225925009352817453ex_rat(G2,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_6402_prod_Oremove,axiom,
    ! [A4: set_int,X: int,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(X),A4))
       => ( groups1072433553688619179nt_rat(G2,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(G2,X)),groups1072433553688619179nt_rat(G2,minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_6403_prod_Oremove,axiom,
    ! [A4: set_real,X: real,G2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(member_real(X),A4))
       => ( groups4061424788464935467al_rat(G2,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(G2,X)),groups4061424788464935467al_rat(G2,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_6404_prod_Oremove,axiom,
    ! [A4: set_nat,X: nat,G2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(member_nat(X),A4))
       => ( groups73079841787564623at_rat(G2,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G2,X)),groups73079841787564623at_rat(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X,bot_bot_set_nat)))) ) ) ) ).

% prod.remove
tff(fact_6405_prod_Oremove,axiom,
    ! [A4: set_complex,X: complex,G2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(member_complex(X),A4))
       => ( groups861055069439313189ex_nat(G2,A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_complex_nat(G2,X)),groups861055069439313189ex_nat(G2,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_6406_prod_Oremove,axiom,
    ! [A4: set_int,X: int,G2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(member_int(X),A4))
       => ( groups1707563613775114915nt_nat(G2,A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(G2,X)),groups1707563613775114915nt_nat(G2,minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_6407_prod_Oinsert__remove,axiom,
    ! [A4: set_complex,G2: fun_complex_real,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups766887009212190081x_real(G2,insert_complex(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G2,X)),groups766887009212190081x_real(G2,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_6408_prod_Oinsert__remove,axiom,
    ! [A4: set_int,G2: fun_int_real,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups2316167850115554303t_real(G2,insert_int(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G2,X)),groups2316167850115554303t_real(G2,minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_6409_prod_Oinsert__remove,axiom,
    ! [A4: set_real,G2: fun_real_real,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1681761925125756287l_real(G2,insert_real(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G2,X)),groups1681761925125756287l_real(G2,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ).

% prod.insert_remove
tff(fact_6410_prod_Oinsert__remove,axiom,
    ! [A4: set_nat,G2: fun_nat_real,X: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( aa_set_nat_real(groups129246275422532515t_real(G2),insert_nat(X,A4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G2,X)),aa_set_nat_real(groups129246275422532515t_real(G2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X,bot_bot_set_nat)))) ) ) ).

% prod.insert_remove
tff(fact_6411_prod_Oinsert__remove,axiom,
    ! [A4: set_complex,G2: fun_complex_rat,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups225925009352817453ex_rat(G2,insert_complex(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_complex_rat(G2,X)),groups225925009352817453ex_rat(G2,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_6412_prod_Oinsert__remove,axiom,
    ! [A4: set_int,G2: fun_int_rat,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups1072433553688619179nt_rat(G2,insert_int(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(G2,X)),groups1072433553688619179nt_rat(G2,minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_6413_prod_Oinsert__remove,axiom,
    ! [A4: set_real,G2: fun_real_rat,X: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups4061424788464935467al_rat(G2,insert_real(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(G2,X)),groups4061424788464935467al_rat(G2,minus_minus_set_real(A4,insert_real(X,bot_bot_set_real)))) ) ) ).

% prod.insert_remove
tff(fact_6414_prod_Oinsert__remove,axiom,
    ! [A4: set_nat,G2: fun_nat_rat,X: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( groups73079841787564623at_rat(G2,insert_nat(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G2,X)),groups73079841787564623at_rat(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X,bot_bot_set_nat)))) ) ) ).

% prod.insert_remove
tff(fact_6415_prod_Oinsert__remove,axiom,
    ! [A4: set_complex,G2: fun_complex_nat,X: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups861055069439313189ex_nat(G2,insert_complex(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_complex_nat(G2,X)),groups861055069439313189ex_nat(G2,minus_811609699411566653omplex(A4,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_6416_prod_Oinsert__remove,axiom,
    ! [A4: set_int,G2: fun_int_nat,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups1707563613775114915nt_nat(G2,insert_int(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(G2,X)),groups1707563613775114915nt_nat(G2,minus_minus_set_int(A4,insert_int(X,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_6417_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),inf_inf_set_int(A4,B5)))
             => ( aa_int_Code_integer(G2,X3) = one_one_Code_integer ) )
         => ( groups3827104343326376752nteger(G2,sup_sup_set_int(A4,B5)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(G2,A4)),groups3827104343326376752nteger(G2,B5)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6418_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_co8153928105989009288nteger] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),inf_inf_set_complex(A4,B5)))
             => ( aa_com4217951221296830281nteger(G2,X3) = one_one_Code_integer ) )
         => ( groups8682486955453173170nteger(G2,sup_sup_set_complex(A4,B5)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups8682486955453173170nteger(G2,A4)),groups8682486955453173170nteger(G2,B5)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6419_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),inf_inf_set_int(A4,B5)))
             => ( aa_int_complex(G2,X3) = one_one_complex ) )
         => ( groups7440179247065528705omplex(G2,sup_sup_set_int(A4,B5)) = aa_complex_complex(times_times_complex(groups7440179247065528705omplex(G2,A4)),groups7440179247065528705omplex(G2,B5)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6420_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),inf_inf_set_complex(A4,B5)))
             => ( aa_complex_complex(G2,X3) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G2,sup_sup_set_complex(A4,B5)) = aa_complex_complex(times_times_complex(groups3708469109370488835omplex(G2,A4)),groups3708469109370488835omplex(G2,B5)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6421_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B5: set_nat,G2: fun_nat_Code_integer] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5)))
             => ( aa_nat_Code_integer(G2,X3) = one_one_Code_integer ) )
         => ( groups3455450783089532116nteger(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3455450783089532116nteger(G2,A4)),groups3455450783089532116nteger(G2,B5)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6422_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B5: set_nat,G2: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5)))
             => ( aa_nat_complex(G2,X3) = one_one_complex ) )
         => ( aa_set_nat_complex(groups6464643781859351333omplex(G2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5)) = aa_complex_complex(times_times_complex(aa_set_nat_complex(groups6464643781859351333omplex(G2),A4)),aa_set_nat_complex(groups6464643781859351333omplex(G2),B5)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6423_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),inf_inf_set_int(A4,B5)))
             => ( aa_int_real(G2,X3) = one_one_real ) )
         => ( groups2316167850115554303t_real(G2,sup_sup_set_int(A4,B5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups2316167850115554303t_real(G2,A4)),groups2316167850115554303t_real(G2,B5)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6424_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),inf_inf_set_complex(A4,B5)))
             => ( aa_complex_real(G2,X3) = one_one_real ) )
         => ( groups766887009212190081x_real(G2,sup_sup_set_complex(A4,B5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(G2,A4)),groups766887009212190081x_real(G2,B5)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6425_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B5: set_nat,G2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5)))
             => ( aa_nat_real(G2,X3) = one_one_real ) )
         => ( aa_set_nat_real(groups129246275422532515t_real(G2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G2),A4)),aa_set_nat_real(groups129246275422532515t_real(G2),B5)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6426_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),inf_inf_set_int(A4,B5)))
             => ( aa_int_rat(G2,X3) = one_one_rat ) )
         => ( groups1072433553688619179nt_rat(G2,sup_sup_set_int(A4,B5)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G2,A4)),groups1072433553688619179nt_rat(G2,B5)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6427_prod_Ounion__disjoint,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( ( inf_inf_set_complex(A4,B5) = bot_bot_set_complex )
         => ( groups766887009212190081x_real(G2,sup_sup_set_complex(A4,B5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(G2,A4)),groups766887009212190081x_real(G2,B5)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_6428_prod_Ounion__disjoint,axiom,
    ! [A4: set_nat,B5: set_nat,G2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5) = bot_bot_set_nat )
         => ( aa_set_nat_real(groups129246275422532515t_real(G2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G2),A4)),aa_set_nat_real(groups129246275422532515t_real(G2),B5)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_6429_prod_Ounion__disjoint,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( ( inf_inf_set_int(A4,B5) = bot_bot_set_int )
         => ( groups2316167850115554303t_real(G2,sup_sup_set_int(A4,B5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups2316167850115554303t_real(G2,A4)),groups2316167850115554303t_real(G2,B5)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_6430_prod_Ounion__disjoint,axiom,
    ! [A4: set_real,B5: set_real,G2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(finite_finite_real,B5))
       => ( ( inf_inf_set_real(A4,B5) = bot_bot_set_real )
         => ( groups1681761925125756287l_real(G2,sup_sup_set_real(A4,B5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups1681761925125756287l_real(G2,A4)),groups1681761925125756287l_real(G2,B5)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_6431_prod_Ounion__disjoint,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( ( inf_inf_set_complex(A4,B5) = bot_bot_set_complex )
         => ( groups225925009352817453ex_rat(G2,sup_sup_set_complex(A4,B5)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups225925009352817453ex_rat(G2,A4)),groups225925009352817453ex_rat(G2,B5)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_6432_prod_Ounion__disjoint,axiom,
    ! [A4: set_nat,B5: set_nat,G2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5) = bot_bot_set_nat )
         => ( groups73079841787564623at_rat(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G2,A4)),groups73079841787564623at_rat(G2,B5)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_6433_prod_Ounion__disjoint,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( ( inf_inf_set_int(A4,B5) = bot_bot_set_int )
         => ( groups1072433553688619179nt_rat(G2,sup_sup_set_int(A4,B5)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G2,A4)),groups1072433553688619179nt_rat(G2,B5)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_6434_prod_Ounion__disjoint,axiom,
    ! [A4: set_real,B5: set_real,G2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( pp(aa_set_real_bool(finite_finite_real,B5))
       => ( ( inf_inf_set_real(A4,B5) = bot_bot_set_real )
         => ( groups4061424788464935467al_rat(G2,sup_sup_set_real(A4,B5)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups4061424788464935467al_rat(G2,A4)),groups4061424788464935467al_rat(G2,B5)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_6435_prod_Ounion__disjoint,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( ( inf_inf_set_complex(A4,B5) = bot_bot_set_complex )
         => ( groups861055069439313189ex_nat(G2,sup_sup_set_complex(A4,B5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G2,A4)),groups861055069439313189ex_nat(G2,B5)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_6436_prod_Ounion__disjoint,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( ( inf_inf_set_int(A4,B5) = bot_bot_set_int )
         => ( groups1707563613775114915nt_nat(G2,sup_sup_set_int(A4,B5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G2,A4)),groups1707563613775114915nt_nat(G2,B5)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_6437_binomial__maximum_H,axiom,
    ! [N: nat,K2: 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))),N)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N))) ).

% binomial_maximum'
tff(fact_6438_binomial__mono,axiom,
    ! [K2: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),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)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6))) ) ) ).

% binomial_mono
tff(fact_6439_binomial__antimono,axiom,
    ! [K2: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2))) ) ) ) ).

% binomial_antimono
tff(fact_6440_binomial__maximum,axiom,
    ! [N: nat,K2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% binomial_maximum
tff(fact_6441_prod_Ounion__diff2,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( groups2316167850115554303t_real(G2,sup_sup_set_int(A4,B5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups2316167850115554303t_real(G2,minus_minus_set_int(A4,B5))),groups2316167850115554303t_real(G2,minus_minus_set_int(B5,A4)))),groups2316167850115554303t_real(G2,inf_inf_set_int(A4,B5))) ) ) ) ).

% prod.union_diff2
tff(fact_6442_prod_Ounion__diff2,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( groups766887009212190081x_real(G2,sup_sup_set_complex(A4,B5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(G2,minus_811609699411566653omplex(A4,B5))),groups766887009212190081x_real(G2,minus_811609699411566653omplex(B5,A4)))),groups766887009212190081x_real(G2,inf_inf_set_complex(A4,B5))) ) ) ) ).

% prod.union_diff2
tff(fact_6443_prod_Ounion__diff2,axiom,
    ! [A4: set_nat,B5: set_nat,G2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( aa_set_nat_real(groups129246275422532515t_real(G2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B5))),aa_set_nat_real(groups129246275422532515t_real(G2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B5),A4)))),aa_set_nat_real(groups129246275422532515t_real(G2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5))) ) ) ) ).

% prod.union_diff2
tff(fact_6444_prod_Ounion__diff2,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( groups1072433553688619179nt_rat(G2,sup_sup_set_int(A4,B5)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G2,minus_minus_set_int(A4,B5))),groups1072433553688619179nt_rat(G2,minus_minus_set_int(B5,A4)))),groups1072433553688619179nt_rat(G2,inf_inf_set_int(A4,B5))) ) ) ) ).

% prod.union_diff2
tff(fact_6445_prod_Ounion__diff2,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( groups225925009352817453ex_rat(G2,sup_sup_set_complex(A4,B5)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups225925009352817453ex_rat(G2,minus_811609699411566653omplex(A4,B5))),groups225925009352817453ex_rat(G2,minus_811609699411566653omplex(B5,A4)))),groups225925009352817453ex_rat(G2,inf_inf_set_complex(A4,B5))) ) ) ) ).

% prod.union_diff2
tff(fact_6446_prod_Ounion__diff2,axiom,
    ! [A4: set_nat,B5: set_nat,G2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( groups73079841787564623at_rat(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B5))),groups73079841787564623at_rat(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B5),A4)))),groups73079841787564623at_rat(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5))) ) ) ) ).

% prod.union_diff2
tff(fact_6447_prod_Ounion__diff2,axiom,
    ! [A4: set_int,B5: set_int,G2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( groups1707563613775114915nt_nat(G2,sup_sup_set_int(A4,B5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G2,minus_minus_set_int(A4,B5))),groups1707563613775114915nt_nat(G2,minus_minus_set_int(B5,A4)))),groups1707563613775114915nt_nat(G2,inf_inf_set_int(A4,B5))) ) ) ) ).

% prod.union_diff2
tff(fact_6448_prod_Ounion__diff2,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( groups861055069439313189ex_nat(G2,sup_sup_set_complex(A4,B5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G2,minus_811609699411566653omplex(A4,B5))),groups861055069439313189ex_nat(G2,minus_811609699411566653omplex(B5,A4)))),groups861055069439313189ex_nat(G2,inf_inf_set_complex(A4,B5))) ) ) ) ).

% prod.union_diff2
tff(fact_6449_prod_Ounion__diff2,axiom,
    ! [A4: set_complex,B5: set_complex,G2: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( groups858564598930262913ex_int(G2,sup_sup_set_complex(A4,B5)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups858564598930262913ex_int(G2,minus_811609699411566653omplex(A4,B5))),groups858564598930262913ex_int(G2,minus_811609699411566653omplex(B5,A4)))),groups858564598930262913ex_int(G2,inf_inf_set_complex(A4,B5))) ) ) ) ).

% prod.union_diff2
tff(fact_6450_prod_Ounion__diff2,axiom,
    ! [A4: set_nat,B5: set_nat,G2: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( groups705719431365010083at_int(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B5))),groups705719431365010083at_int(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B5),A4)))),groups705719431365010083at_int(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5))) ) ) ) ).

% prod.union_diff2
tff(fact_6451_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_real,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_6452_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_rat,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups73079841787564623at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),groups73079841787564623at_rat(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_6453_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_int,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),groups705719431365010083at_int(G2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_6454_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: fun_nat_nat,P: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P)))) ) ) ).

% prod.ub_add_nat
tff(fact_6455_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [B: nat,A: nat,F2: 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(F2,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(F2,A),B),Acc2) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F2,A),Acc2)) ) ) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_6456_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa2: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa2),Xb),Xc) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa2))
         => ( Y = Xc ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),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)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa2),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_6457_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real,C2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_mb(A),B),C2),S3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(B,A)),groups766887009212190081x_real(C2,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_mb(A),B),C2),S3) = groups766887009212190081x_real(C2,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6458_prod_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: fun_int_real,C2: 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(aa_fun6524208927192123834t_real(aTP_Lamm_mc(A),B),C2),S3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(B,A)),groups2316167850115554303t_real(C2,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_mc(A),B),C2),S3) = groups2316167850115554303t_real(C2,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6459_prod_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: fun_real_real,C2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( pp(aa_set_real_bool(member_real(A),S3))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_md(A),B),C2),S3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(B,A)),groups1681761925125756287l_real(C2,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S3))
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_md(A),B),C2),S3) = groups1681761925125756287l_real(C2,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6460_prod_Odelta__remove,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_real,C2: 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(aa_fun8674848580393814182t_real(aTP_Lamm_me(A),B),C2)),S3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(B,A)),aa_set_nat_real(groups129246275422532515t_real(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S3),insert_nat(A,bot_bot_set_nat)))) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_me(A),B),C2)),S3) = aa_set_nat_real(groups129246275422532515t_real(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S3),insert_nat(A,bot_bot_set_nat))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6461_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_rat,C2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups225925009352817453ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_mf(A),B),C2),S3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_complex_rat(B,A)),groups225925009352817453ex_rat(C2,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups225925009352817453ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_mf(A),B),C2),S3) = groups225925009352817453ex_rat(C2,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6462_prod_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: fun_int_rat,C2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_mg(A),B),C2),S3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(B,A)),groups1072433553688619179nt_rat(C2,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_mg(A),B),C2),S3) = groups1072433553688619179nt_rat(C2,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6463_prod_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat,C2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( pp(aa_set_real_bool(member_real(A),S3))
         => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_mh(A),B),C2),S3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(B,A)),groups4061424788464935467al_rat(C2,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S3))
         => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_mh(A),B),C2),S3) = groups4061424788464935467al_rat(C2,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6464_prod_Odelta__remove,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_rat,C2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_mi(A),B),C2),S3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(B,A)),groups73079841787564623at_rat(C2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S3),insert_nat(A,bot_bot_set_nat)))) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_mi(A),B),C2),S3) = groups73079841787564623at_rat(C2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S3),insert_nat(A,bot_bot_set_nat))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6465_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_nat,C2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups861055069439313189ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_mj(A),B),C2),S3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_complex_nat(B,A)),groups861055069439313189ex_nat(C2,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups861055069439313189ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_mj(A),B),C2),S3) = groups861055069439313189ex_nat(C2,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6466_prod_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: fun_int_nat,C2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_mk(A),B),C2),S3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(B,A)),groups1707563613775114915nt_nat(C2,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_mk(A),B),C2),S3) = groups1707563613775114915nt_nat(C2,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int))) ) ) ) ) ).

% prod.delta_remove
tff(fact_6467_power__diff__1__eq,axiom,
    ! [X: code_integer,N: nat] : minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),N),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(X,one_one_Code_integer)),groups7501900531339628137nteger(power_8256067586552552935nteger(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6468_power__diff__1__eq,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),one_one_complex)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6469_power__diff__1__eq,axiom,
    ! [X: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),N)),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),one_one_rat)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6470_power__diff__1__eq,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),N)),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,N))) ).

% power_diff_1_eq
tff(fact_6471_power__diff__1__eq,axiom,
    ! [X: real,N: nat] : aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(X),one_one_real)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_6472_one__diff__power__eq,axiom,
    ! [X: code_integer,N: nat] : minus_8373710615458151222nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(one_one_Code_integer,X)),groups7501900531339628137nteger(power_8256067586552552935nteger(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6473_one__diff__power__eq,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6474_one__diff__power__eq,axiom,
    ! [X: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6475_one__diff__power__eq,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),N)) = 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,N))) ).

% one_diff_power_eq
tff(fact_6476_one__diff__power__eq,axiom,
    ! [X: real,N: nat] : aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_6477_geometric__sum,axiom,
    ! [X: complex,N: nat] :
      ( ( X != one_one_complex )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_6478_geometric__sum,axiom,
    ! [X: rat,N: nat] :
      ( ( X != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),N)),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),one_one_rat)) ) ) ).

% geometric_sum
tff(fact_6479_geometric__sum,axiom,
    ! [X: real,N: nat] :
      ( ( X != one_one_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),one_one_real)),aa_real_real(minus_minus_real(X),one_one_real)) ) ) ).

% geometric_sum
tff(fact_6480_prod__mono2,axiom,
    ! [B5: set_real,A4: set_real,F2: fun_re6978948838062957830nteger] :
      ( pp(aa_set_real_bool(finite_finite_real,B5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B5))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B5,A4)))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_real_Code_integer(F2,B3))) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),A4))
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(F2,A3))) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups6225526099057966256nteger(F2,A4)),groups6225526099057966256nteger(F2,B5))) ) ) ) ) ).

% prod_mono2
tff(fact_6481_prod__mono2,axiom,
    ! [B5: set_int,A4: set_int,F2: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,B5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B5))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B5,A4)))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(F2,B3))) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),A4))
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F2,A3))) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups3827104343326376752nteger(F2,A4)),groups3827104343326376752nteger(F2,B5))) ) ) ) ) ).

% prod_mono2
tff(fact_6482_prod__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F2: fun_co8153928105989009288nteger] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B5))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B5,A4)))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_com4217951221296830281nteger(F2,B3))) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),A4))
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(F2,A3))) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups8682486955453173170nteger(F2,A4)),groups8682486955453173170nteger(F2,B5))) ) ) ) ) ).

% prod_mono2
tff(fact_6483_prod__mono2,axiom,
    ! [B5: set_real,A4: set_real,F2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B5))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B5,A4)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F2,B3))) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),A4))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F2,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F2,A4)),groups1681761925125756287l_real(F2,B5))) ) ) ) ) ).

% prod_mono2
tff(fact_6484_prod__mono2,axiom,
    ! [B5: set_int,A4: set_int,F2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,B5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B5))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B5,A4)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F2,B3))) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),A4))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F2,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F2,A4)),groups2316167850115554303t_real(F2,B5))) ) ) ) ) ).

% prod_mono2
tff(fact_6485_prod__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B5))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B5,A4)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F2,B3))) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),A4))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F2,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups766887009212190081x_real(F2,A4)),groups766887009212190081x_real(F2,B5))) ) ) ) ) ).

% prod_mono2
tff(fact_6486_prod__mono2,axiom,
    ! [B5: set_real,A4: set_real,F2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B5))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B5,A4)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F2,B3))) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),A4))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F2,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F2,A4)),groups4061424788464935467al_rat(F2,B5))) ) ) ) ) ).

% prod_mono2
tff(fact_6487_prod__mono2,axiom,
    ! [B5: set_int,A4: set_int,F2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,B5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A4),B5))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B5,A4)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(F2,B3))) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),A4))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F2,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1072433553688619179nt_rat(F2,A4)),groups1072433553688619179nt_rat(F2,B5))) ) ) ) ) ).

% prod_mono2
tff(fact_6488_prod__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A4),B5))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B5,A4)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_complex_rat(F2,B3))) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),A4))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F2,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups225925009352817453ex_rat(F2,A4)),groups225925009352817453ex_rat(F2,B5))) ) ) ) ) ).

% prod_mono2
tff(fact_6489_prod__mono2,axiom,
    ! [B5: set_real,A4: set_real,F2: fun_real_int] :
      ( pp(aa_set_real_bool(finite_finite_real,B5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A4),B5))
       => ( ! [B3: real] :
              ( pp(aa_set_real_bool(member_real(B3),minus_minus_set_real(B5,A4)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(F2,B3))) )
         => ( ! [A3: real] :
                ( pp(aa_set_real_bool(member_real(A3),A4))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F2,A3))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4694064378042380927al_int(F2,A4)),groups4694064378042380927al_int(F2,B5))) ) ) ) ) ).

% prod_mono2
tff(fact_6490_prod__le__power,axiom,
    ! [A4: set_real,F2: fun_re6978948838062957830nteger,N: code_integer,K2: nat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(F2,I3)))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_real_Code_integer(F2,I3)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(A4)),K2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),N))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups6225526099057966256nteger(F2,A4)),aa_nat_Code_integer(power_8256067586552552935nteger(N),K2))) ) ) ) ).

% prod_le_power
tff(fact_6491_prod__le__power,axiom,
    ! [A4: set_nat,F2: fun_nat_Code_integer,N: code_integer,K2: nat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A4))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F2,I3)))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(F2,I3)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),K2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),N))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups3455450783089532116nteger(F2,A4)),aa_nat_Code_integer(power_8256067586552552935nteger(N),K2))) ) ) ) ).

% prod_le_power
tff(fact_6492_prod__le__power,axiom,
    ! [A4: set_complex,F2: fun_co8153928105989009288nteger,N: code_integer,K2: nat] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A4))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(F2,I3)))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_com4217951221296830281nteger(F2,I3)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A4)),K2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),N))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups8682486955453173170nteger(F2,A4)),aa_nat_Code_integer(power_8256067586552552935nteger(N),K2))) ) ) ) ).

% prod_le_power
tff(fact_6493_prod__le__power,axiom,
    ! [A4: set_int,F2: fun_int_Code_integer,N: code_integer,K2: nat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F2,I3)))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(F2,I3)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(A4)),K2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),N))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups3827104343326376752nteger(F2,A4)),aa_nat_Code_integer(power_8256067586552552935nteger(N),K2))) ) ) ) ).

% prod_le_power
tff(fact_6494_prod__le__power,axiom,
    ! [A4: set_real,F2: fun_real_real,N: real,K2: nat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F2,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F2,I3)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(A4)),K2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F2,A4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,N),K2))) ) ) ) ).

% prod_le_power
tff(fact_6495_prod__le__power,axiom,
    ! [A4: set_nat,F2: fun_nat_real,N: real,K2: nat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F2,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F2,I3)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),K2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F2),A4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,N),K2))) ) ) ) ).

% prod_le_power
tff(fact_6496_prod__le__power,axiom,
    ! [A4: set_complex,F2: fun_complex_real,N: real,K2: nat] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F2,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F2,I3)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(A4)),K2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups766887009212190081x_real(F2,A4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,N),K2))) ) ) ) ).

% prod_le_power
tff(fact_6497_prod__le__power,axiom,
    ! [A4: set_int,F2: fun_int_real,N: real,K2: nat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),A4))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F2,I3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F2,I3)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_int(A4)),K2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F2,A4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,N),K2))) ) ) ) ).

% prod_le_power
tff(fact_6498_prod__le__power,axiom,
    ! [A4: set_real,F2: fun_real_rat,N: rat,K2: nat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),A4))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F2,I3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F2,I3)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(A4)),K2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F2,A4)),aa_nat_rat(power_power_rat(N),K2))) ) ) ) ).

% prod_le_power
tff(fact_6499_prod__le__power,axiom,
    ! [A4: set_nat,F2: fun_nat_rat,N: rat,K2: nat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),A4))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F2,I3)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,I3)),N)) ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),K2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups73079841787564623at_rat(F2,A4)),aa_nat_rat(power_power_rat(N),K2))) ) ) ) ).

% prod_le_power
tff(fact_6500_prod__Un,axiom,
    ! [A4: set_int,B5: set_int,F2: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),inf_inf_set_int(A4,B5)))
             => ( aa_int_complex(F2,X3) != zero_zero_complex ) )
         => ( groups7440179247065528705omplex(F2,sup_sup_set_int(A4,B5)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(groups7440179247065528705omplex(F2,A4)),groups7440179247065528705omplex(F2,B5))),groups7440179247065528705omplex(F2,inf_inf_set_int(A4,B5))) ) ) ) ) ).

% prod_Un
tff(fact_6501_prod__Un,axiom,
    ! [A4: set_complex,B5: set_complex,F2: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),inf_inf_set_complex(A4,B5)))
             => ( aa_complex_complex(F2,X3) != zero_zero_complex ) )
         => ( groups3708469109370488835omplex(F2,sup_sup_set_complex(A4,B5)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(groups3708469109370488835omplex(F2,A4)),groups3708469109370488835omplex(F2,B5))),groups3708469109370488835omplex(F2,inf_inf_set_complex(A4,B5))) ) ) ) ) ).

% prod_Un
tff(fact_6502_prod__Un,axiom,
    ! [A4: set_nat,B5: set_nat,F2: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5)))
             => ( aa_nat_complex(F2,X3) != zero_zero_complex ) )
         => ( aa_set_nat_complex(groups6464643781859351333omplex(F2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_set_nat_complex(groups6464643781859351333omplex(F2),A4)),aa_set_nat_complex(groups6464643781859351333omplex(F2),B5))),aa_set_nat_complex(groups6464643781859351333omplex(F2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5))) ) ) ) ) ).

% prod_Un
tff(fact_6503_prod__Un,axiom,
    ! [A4: set_int,B5: set_int,F2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),inf_inf_set_int(A4,B5)))
             => ( aa_int_real(F2,X3) != zero_zero_real ) )
         => ( groups2316167850115554303t_real(F2,sup_sup_set_int(A4,B5)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups2316167850115554303t_real(F2,A4)),groups2316167850115554303t_real(F2,B5))),groups2316167850115554303t_real(F2,inf_inf_set_int(A4,B5))) ) ) ) ) ).

% prod_Un
tff(fact_6504_prod__Un,axiom,
    ! [A4: set_complex,B5: set_complex,F2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),inf_inf_set_complex(A4,B5)))
             => ( aa_complex_real(F2,X3) != zero_zero_real ) )
         => ( groups766887009212190081x_real(F2,sup_sup_set_complex(A4,B5)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(F2,A4)),groups766887009212190081x_real(F2,B5))),groups766887009212190081x_real(F2,inf_inf_set_complex(A4,B5))) ) ) ) ) ).

% prod_Un
tff(fact_6505_prod__Un,axiom,
    ! [A4: set_nat,B5: set_nat,F2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5)))
             => ( aa_nat_real(F2,X3) != zero_zero_real ) )
         => ( aa_set_nat_real(groups129246275422532515t_real(F2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(F2),A4)),aa_set_nat_real(groups129246275422532515t_real(F2),B5))),aa_set_nat_real(groups129246275422532515t_real(F2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5))) ) ) ) ) ).

% prod_Un
tff(fact_6506_prod__Un,axiom,
    ! [A4: set_int,B5: set_int,F2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_set_int_bool(finite_finite_int,B5))
       => ( ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),inf_inf_set_int(A4,B5)))
             => ( aa_int_rat(F2,X3) != zero_zero_rat ) )
         => ( groups1072433553688619179nt_rat(F2,sup_sup_set_int(A4,B5)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(F2,A4)),groups1072433553688619179nt_rat(F2,B5))),groups1072433553688619179nt_rat(F2,inf_inf_set_int(A4,B5))) ) ) ) ) ).

% prod_Un
tff(fact_6507_prod__Un,axiom,
    ! [A4: set_complex,B5: set_complex,F2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B5))
       => ( ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),inf_inf_set_complex(A4,B5)))
             => ( aa_complex_rat(F2,X3) != zero_zero_rat ) )
         => ( groups225925009352817453ex_rat(F2,sup_sup_set_complex(A4,B5)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups225925009352817453ex_rat(F2,A4)),groups225925009352817453ex_rat(F2,B5))),groups225925009352817453ex_rat(F2,inf_inf_set_complex(A4,B5))) ) ) ) ) ).

% prod_Un
tff(fact_6508_prod__Un,axiom,
    ! [A4: set_nat,B5: set_nat,F2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B5))
       => ( ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5)))
             => ( aa_nat_rat(F2,X3) != zero_zero_rat ) )
         => ( groups73079841787564623at_rat(F2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B5)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(F2,A4)),groups73079841787564623at_rat(F2,B5))),groups73079841787564623at_rat(F2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B5))) ) ) ) ) ).

% prod_Un
tff(fact_6509_prod__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F2: fun_Pr4050773816208733250omplex] :
      ( finite6177210948735845034at_nat(A4)
     => ( finite6177210948735845034at_nat(B5)
       => ( ! [X3: product_prod_nat_nat] :
              ( member8440522571783428010at_nat(X3,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B5))
             => ( aa_Pro82899191565270131omplex(F2,X3) != zero_zero_complex ) )
         => ( groups8110221916422527690omplex(F2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B5)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(groups8110221916422527690omplex(F2,A4)),groups8110221916422527690omplex(F2,B5))),groups8110221916422527690omplex(F2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B5))) ) ) ) ) ).

% prod_Un
tff(fact_6510_prod__diff1,axiom,
    ! [A4: set_complex,F2: fun_complex_complex,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( aa_complex_complex(F2,A) != zero_zero_complex )
       => ( ( pp(aa_set_complex_bool(member_complex(A),A4))
           => ( groups3708469109370488835omplex(F2,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = aa_complex_complex(divide1717551699836669952omplex(groups3708469109370488835omplex(F2,A4)),aa_complex_complex(F2,A)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(A),A4))
           => ( groups3708469109370488835omplex(F2,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = groups3708469109370488835omplex(F2,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_6511_prod__diff1,axiom,
    ! [A4: set_int,F2: fun_int_complex,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( aa_int_complex(F2,A) != zero_zero_complex )
       => ( ( pp(aa_set_int_bool(member_int(A),A4))
           => ( groups7440179247065528705omplex(F2,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = aa_complex_complex(divide1717551699836669952omplex(groups7440179247065528705omplex(F2,A4)),aa_int_complex(F2,A)) ) )
          & ( ~ pp(aa_set_int_bool(member_int(A),A4))
           => ( groups7440179247065528705omplex(F2,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = groups7440179247065528705omplex(F2,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_6512_prod__diff1,axiom,
    ! [A4: set_real,F2: fun_real_complex,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( aa_real_complex(F2,A) != zero_zero_complex )
       => ( ( pp(aa_set_real_bool(member_real(A),A4))
           => ( groups713298508707869441omplex(F2,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = aa_complex_complex(divide1717551699836669952omplex(groups713298508707869441omplex(F2,A4)),aa_real_complex(F2,A)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(A),A4))
           => ( groups713298508707869441omplex(F2,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = groups713298508707869441omplex(F2,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_6513_prod__diff1,axiom,
    ! [A4: set_nat,F2: fun_nat_complex,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( aa_nat_complex(F2,A) != zero_zero_complex )
       => ( ( pp(aa_set_nat_bool(member_nat(A),A4))
           => ( aa_set_nat_complex(groups6464643781859351333omplex(F2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,bot_bot_set_nat))) = aa_complex_complex(divide1717551699836669952omplex(aa_set_nat_complex(groups6464643781859351333omplex(F2),A4)),aa_nat_complex(F2,A)) ) )
          & ( ~ pp(aa_set_nat_bool(member_nat(A),A4))
           => ( aa_set_nat_complex(groups6464643781859351333omplex(F2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,bot_bot_set_nat))) = aa_set_nat_complex(groups6464643781859351333omplex(F2),A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_6514_prod__diff1,axiom,
    ! [A4: set_complex,F2: fun_complex_real,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( aa_complex_real(F2,A) != zero_zero_real )
       => ( ( pp(aa_set_complex_bool(member_complex(A),A4))
           => ( groups766887009212190081x_real(F2,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = aa_real_real(divide_divide_real(groups766887009212190081x_real(F2,A4)),aa_complex_real(F2,A)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(A),A4))
           => ( groups766887009212190081x_real(F2,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = groups766887009212190081x_real(F2,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_6515_prod__diff1,axiom,
    ! [A4: set_int,F2: fun_int_real,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( aa_int_real(F2,A) != zero_zero_real )
       => ( ( pp(aa_set_int_bool(member_int(A),A4))
           => ( groups2316167850115554303t_real(F2,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = aa_real_real(divide_divide_real(groups2316167850115554303t_real(F2,A4)),aa_int_real(F2,A)) ) )
          & ( ~ pp(aa_set_int_bool(member_int(A),A4))
           => ( groups2316167850115554303t_real(F2,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = groups2316167850115554303t_real(F2,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_6516_prod__diff1,axiom,
    ! [A4: set_real,F2: fun_real_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( ( aa_real_real(F2,A) != zero_zero_real )
       => ( ( pp(aa_set_real_bool(member_real(A),A4))
           => ( groups1681761925125756287l_real(F2,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = aa_real_real(divide_divide_real(groups1681761925125756287l_real(F2,A4)),aa_real_real(F2,A)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(A),A4))
           => ( groups1681761925125756287l_real(F2,minus_minus_set_real(A4,insert_real(A,bot_bot_set_real))) = groups1681761925125756287l_real(F2,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_6517_prod__diff1,axiom,
    ! [A4: set_nat,F2: fun_nat_real,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( aa_nat_real(F2,A) != zero_zero_real )
       => ( ( pp(aa_set_nat_bool(member_nat(A),A4))
           => ( aa_set_nat_real(groups129246275422532515t_real(F2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,bot_bot_set_nat))) = aa_real_real(divide_divide_real(aa_set_nat_real(groups129246275422532515t_real(F2),A4)),aa_nat_real(F2,A)) ) )
          & ( ~ pp(aa_set_nat_bool(member_nat(A),A4))
           => ( aa_set_nat_real(groups129246275422532515t_real(F2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,bot_bot_set_nat))) = aa_set_nat_real(groups129246275422532515t_real(F2),A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_6518_prod__diff1,axiom,
    ! [A4: set_complex,F2: fun_complex_rat,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( ( aa_complex_rat(F2,A) != zero_zero_rat )
       => ( ( pp(aa_set_complex_bool(member_complex(A),A4))
           => ( groups225925009352817453ex_rat(F2,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,groups225925009352817453ex_rat(F2,A4)),aa_complex_rat(F2,A)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(A),A4))
           => ( groups225925009352817453ex_rat(F2,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = groups225925009352817453ex_rat(F2,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_6519_prod__diff1,axiom,
    ! [A4: set_int,F2: fun_int_rat,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( ( aa_int_rat(F2,A) != zero_zero_rat )
       => ( ( pp(aa_set_int_bool(member_int(A),A4))
           => ( groups1072433553688619179nt_rat(F2,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,groups1072433553688619179nt_rat(F2,A4)),aa_int_rat(F2,A)) ) )
          & ( ~ pp(aa_set_int_bool(member_int(A),A4))
           => ( groups1072433553688619179nt_rat(F2,minus_minus_set_int(A4,insert_int(A,bot_bot_set_int))) = groups1072433553688619179nt_rat(F2,A4) ) ) ) ) ) ).

% prod_diff1
tff(fact_6520_binomial__less__binomial__Suc,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),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,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K2)))) ) ).

% binomial_less_binomial_Suc
tff(fact_6521_binomial__strict__antimono,axiom,
    ! [K2: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2))) ) ) ) ).

% binomial_strict_antimono
tff(fact_6522_binomial__strict__mono,axiom,
    ! [K2: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),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)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6))) ) ) ).

% binomial_strict_mono
tff(fact_6523_central__binomial__odd,axiom,
    ! [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))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_6524_prod__gen__delta,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex,C2: complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( pp(aa_set_real_bool(member_real(A),S3))
         => ( groups713298508707869441omplex(aa_com4766338864777664963omplex(aa_fun3608241143110160151omplex(aTP_Lamm_ml(A),B),C2),S3) = aa_complex_complex(times_times_complex(aa_real_complex(B,A)),aa_nat_complex(power_power_complex(C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_real(S3)),one_one_nat))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S3))
         => ( groups713298508707869441omplex(aa_com4766338864777664963omplex(aa_fun3608241143110160151omplex(aTP_Lamm_ml(A),B),C2),S3) = aa_nat_complex(power_power_complex(C2),finite_card_real(S3)) ) ) ) ) ).

% prod_gen_delta
tff(fact_6525_prod__gen__delta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex,C2: complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( aa_set_nat_complex(groups6464643781859351333omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_mm(A),B),C2)),S3) = aa_complex_complex(times_times_complex(aa_nat_complex(B,A)),aa_nat_complex(power_power_complex(C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,S3)),one_one_nat))) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( aa_set_nat_complex(groups6464643781859351333omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_mm(A),B),C2)),S3) = aa_nat_complex(power_power_complex(C2),aa_set_nat_nat(finite_card_nat,S3)) ) ) ) ) ).

% prod_gen_delta
tff(fact_6526_prod__gen__delta,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex,C2: complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups7440179247065528705omplex(aa_com9156536935073639875omplex(aa_fun4786838533949490199omplex(aTP_Lamm_mn(A),B),C2),S3) = aa_complex_complex(times_times_complex(aa_int_complex(B,A)),aa_nat_complex(power_power_complex(C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(S3)),one_one_nat))) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups7440179247065528705omplex(aa_com9156536935073639875omplex(aa_fun4786838533949490199omplex(aTP_Lamm_mn(A),B),C2),S3) = aa_nat_complex(power_power_complex(C2),finite_card_int(S3)) ) ) ) ) ).

% prod_gen_delta
tff(fact_6527_prod__gen__delta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_complex,C2: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups3708469109370488835omplex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_mo(A),B),C2),S3) = aa_complex_complex(times_times_complex(aa_complex_complex(B,A)),aa_nat_complex(power_power_complex(C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(S3)),one_one_nat))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups3708469109370488835omplex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_mo(A),B),C2),S3) = aa_nat_complex(power_power_complex(C2),finite_card_complex(S3)) ) ) ) ) ).

% prod_gen_delta
tff(fact_6528_prod__gen__delta,axiom,
    ! [S3: set_real,A: real,B: fun_real_real,C2: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( pp(aa_set_real_bool(member_real(A),S3))
         => ( groups1681761925125756287l_real(aa_rea7340605660300375871l_real(aa_fun4910002773489250577l_real(aTP_Lamm_mp(A),B),C2),S3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(B,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_real(S3)),one_one_nat))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S3))
         => ( groups1681761925125756287l_real(aa_rea7340605660300375871l_real(aa_fun4910002773489250577l_real(aTP_Lamm_mp(A),B),C2),S3) = aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),finite_card_real(S3)) ) ) ) ) ).

% prod_gen_delta
tff(fact_6529_prod__gen__delta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_real,C2: 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_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_mq(A),B),C2)),S3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(B,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,S3)),one_one_nat))) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_mq(A),B),C2)),S3) = aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),aa_set_nat_nat(finite_card_nat,S3)) ) ) ) ) ).

% prod_gen_delta
tff(fact_6530_prod__gen__delta,axiom,
    ! [S3: set_int,A: int,B: fun_int_real,C2: real] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups2316167850115554303t_real(aa_real_fun_int_real(aa_fun8936210173152961553t_real(aTP_Lamm_mr(A),B),C2),S3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(B,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(S3)),one_one_nat))) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups2316167850115554303t_real(aa_real_fun_int_real(aa_fun8936210173152961553t_real(aTP_Lamm_mr(A),B),C2),S3) = aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),finite_card_int(S3)) ) ) ) ) ).

% prod_gen_delta
tff(fact_6531_prod__gen__delta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real,C2: real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups766887009212190081x_real(aa_rea4562580341872191937x_real(aa_fun3210294639341725717x_real(aTP_Lamm_ms(A),B),C2),S3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(B,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(S3)),one_one_nat))) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups766887009212190081x_real(aa_rea4562580341872191937x_real(aa_fun3210294639341725717x_real(aTP_Lamm_ms(A),B),C2),S3) = aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),finite_card_complex(S3)) ) ) ) ) ).

% prod_gen_delta
tff(fact_6532_prod__gen__delta,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat,C2: rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( pp(aa_set_real_bool(member_real(A),S3))
         => ( groups4061424788464935467al_rat(aa_rat_fun_real_rat(aa_fun2328011964990066069al_rat(aTP_Lamm_mt(A),B),C2),S3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(B,A)),aa_nat_rat(power_power_rat(C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_real(S3)),one_one_nat))) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S3))
         => ( groups4061424788464935467al_rat(aa_rat_fun_real_rat(aa_fun2328011964990066069al_rat(aTP_Lamm_mt(A),B),C2),S3) = aa_nat_rat(power_power_rat(C2),finite_card_real(S3)) ) ) ) ) ).

% prod_gen_delta
tff(fact_6533_prod__gen__delta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_rat,C2: rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups73079841787564623at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_mu(A),B),C2),S3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(B,A)),aa_nat_rat(power_power_rat(C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,S3)),one_one_nat))) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups73079841787564623at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_mu(A),B),C2),S3) = aa_nat_rat(power_power_rat(C2),aa_set_nat_nat(finite_card_nat,S3)) ) ) ) ) ).

% prod_gen_delta
tff(fact_6534_pochhammer__Suc__prod,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = groups73079841787564623at_rat(aTP_Lamm_mv(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_6535_pochhammer__Suc__prod,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,N)) = aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_mw,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_6536_pochhammer__Suc__prod,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = groups705719431365010083at_int(aTP_Lamm_mx(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_6537_pochhammer__Suc__prod,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_my,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod
tff(fact_6538_sum__gp__strict,axiom,
    ! [X: complex,N: nat] :
      ( ( ( X = one_one_complex )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_complex(semiri8010041392384452111omplex,N) ) )
      & ( ( X != one_one_complex )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp_strict
tff(fact_6539_sum__gp__strict,axiom,
    ! [X: rat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),N))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)) ) ) ) ).

% sum_gp_strict
tff(fact_6540_sum__gp__strict,axiom,
    ! [X: real,N: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ) )
      & ( ( X != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))),aa_real_real(minus_minus_real(one_one_real),X)) ) ) ) ).

% sum_gp_strict
tff(fact_6541_lemma__termdiff1,axiom,
    ! [Z2: complex,H: complex,M2: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mz(Z2),H),M2)),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_na(Z2),H),M2)),aa_nat_set_nat(set_ord_lessThan_nat,M2)) ).

% lemma_termdiff1
tff(fact_6542_lemma__termdiff1,axiom,
    ! [Z2: rat,H: rat,M2: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nb(Z2),H),M2),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nc(Z2),H),M2),aa_nat_set_nat(set_ord_lessThan_nat,M2)) ).

% lemma_termdiff1
tff(fact_6543_lemma__termdiff1,axiom,
    ! [Z2: int,H: int,M2: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_nd(Z2),H),M2),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ne(Z2),H),M2),aa_nat_set_nat(set_ord_lessThan_nat,M2)) ).

% lemma_termdiff1
tff(fact_6544_lemma__termdiff1,axiom,
    ! [Z2: real,H: real,M2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nf(Z2),H),M2)),aa_nat_set_nat(set_ord_lessThan_nat,M2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ng(Z2),H),M2)),aa_nat_set_nat(set_ord_lessThan_nat,M2)) ).

% lemma_termdiff1
tff(fact_6545_diff__power__eq__sum,axiom,
    ! [X: complex,N: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))),aa_nat_complex(power_power_complex(Y),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nh(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_6546_diff__power__eq__sum,axiom,
    ! [X: rat,N: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ni(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_6547_diff__power__eq__sum,axiom,
    ! [X: int,N: 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,N))),aa_nat_int(power_power_int(Y),aa_nat_nat(suc,N))) = 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_nj(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_6548_diff__power__eq__sum,axiom,
    ! [X: real,N: nat,Y: real] : aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_nk(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_6549_power__diff__sumr2,axiom,
    ! [X: complex,N: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nl(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_6550_power__diff__sumr2,axiom,
    ! [X: rat,N: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(Y),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_nm(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_6551_power__diff__sumr2,axiom,
    ! [X: int,N: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N)) = 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_nn(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_6552_power__diff__sumr2,axiom,
    ! [X: real,N: nat,Y: real] : aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_no(X),N),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_6553_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F2: fun_nat_rat,K5: rat,K2: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F2,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(F2,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6554_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F2: fun_nat_int,K5: int,K2: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F2,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(F2,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6555_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F2: fun_nat_nat,K5: nat,K2: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F2,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(F2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6556_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F2: fun_nat_real,K5: real,K2: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F2,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(F2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6557_prod_Oin__pairs,axiom,
    ! [G2: fun_nat_real,M2: nat,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G2),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))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_np(G2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).

% prod.in_pairs
tff(fact_6558_prod_Oin__pairs,axiom,
    ! [G2: fun_nat_rat,M2: nat,N: nat] : groups73079841787564623at_rat(G2,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))),M2)),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)))) = groups73079841787564623at_rat(aTP_Lamm_nq(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).

% prod.in_pairs
tff(fact_6559_prod_Oin__pairs,axiom,
    ! [G2: fun_nat_int,M2: nat,N: nat] : groups705719431365010083at_int(G2,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))),M2)),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)))) = groups705719431365010083at_int(aTP_Lamm_nr(G2),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).

% prod.in_pairs
tff(fact_6560_prod_Oin__pairs,axiom,
    ! [G2: fun_nat_nat,M2: nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G2),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))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ns(G2)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).

% prod.in_pairs
tff(fact_6561_sum__atLeastAtMost__code,axiom,
    ! [F2: fun_nat_complex,A: nat,B: nat] : aa_set_nat_complex(groups2073611262835488442omplex(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_nt(F2),A,B),zero_zero_complex) ).

% sum_atLeastAtMost_code
tff(fact_6562_sum__atLeastAtMost__code,axiom,
    ! [F2: fun_nat_rat,A: nat,B: nat] : groups2906978787729119204at_rat(F2,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_nu(F2),A,B),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_6563_sum__atLeastAtMost__code,axiom,
    ! [F2: fun_nat_int,A: nat,B: nat] : groups3539618377306564664at_int(F2,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_nv(F2),A,B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_6564_sum__atLeastAtMost__code,axiom,
    ! [F2: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_nw(F2),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_6565_sum__atLeastAtMost__code,axiom,
    ! [F2: fun_nat_real,A: nat,B: nat] : aa_set_nat_real(groups6591440286371151544t_real(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_nx(F2),A),B),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_6566_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_ny(A),N),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_6567_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,N)) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_nz(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_6568_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_oa(A),N),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_6569_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ob(A),N)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% pochhammer_Suc_prod_rev
tff(fact_6570_one__diff__power__eq_H,axiom,
    ! [X: code_integer,N: nat] : minus_8373710615458151222nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(one_one_Code_integer,X)),groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aTP_Lamm_oc(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_6571_one__diff__power__eq_H,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_od(X),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_6572_one__diff__power__eq_H,axiom,
    ! [X: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_oe(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_6573_one__diff__power__eq_H,axiom,
    ! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),N)) = 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_of(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_6574_one__diff__power__eq_H,axiom,
    ! [X: real,N: nat] : aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_og(X),N)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_6575_sum__split__even__odd,axiom,
    ! [F2: fun_nat_real,G2: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_oh(F2),G2)),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))),N))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_oi(F2)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_oj(G2)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum_split_even_odd
tff(fact_6576_zero__less__binomial__iff,axiom,
    ! [N: nat,K2: 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,N),K2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N)) ) ).

% zero_less_binomial_iff
tff(fact_6577_choose__two,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),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,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_6578_binomial__n__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_6579_binomial__Suc__Suc,axiom,
    ! [N: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K2))) ).

% binomial_Suc_Suc
tff(fact_6580_binomial__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2)) ) ).

% binomial_eq_0_iff
tff(fact_6581_binomial__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,zero_zero_nat)) = N ).

% binomial_1
tff(fact_6582_binomial__0__Suc,axiom,
    ! [K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K2)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_6583_binomial__Suc__n,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),N) = aa_nat_nat(suc,N) ).

% binomial_Suc_n
tff(fact_6584_prod__pos__nat__iff,axiom,
    ! [A4: set_int,F2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1707563613775114915nt_nat(F2,A4)))
      <=> ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F2,X4))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6585_prod__pos__nat__iff,axiom,
    ! [A4: set_complex,F2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups861055069439313189ex_nat(F2,A4)))
      <=> ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F2,X4))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6586_prod__pos__nat__iff,axiom,
    ! [A4: set_nat,F2: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F2),A4)))
      <=> ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F2,X4))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_6587_prod__int__eq,axiom,
    ! [I2: nat,J2: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I2),J2)) = groups1705073143266064639nt_int(aTP_Lamm_as,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I2),aa_nat_int(semiri1314217659103216013at_int,J2))) ).

% prod_int_eq
tff(fact_6588_prod__int__plus__eq,axiom,
    ! [I2: nat,J2: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J2))) = groups1705073143266064639nt_int(aTP_Lamm_as,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J2)))) ).

% prod_int_plus_eq
tff(fact_6589_binomial__eq__0,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_6590_Suc__times__binomial,axiom,
    ! [K2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)) ).

% Suc_times_binomial
tff(fact_6591_Suc__times__binomial__eq,axiom,
    ! [N: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K2))),aa_nat_nat(suc,K2)) ).

% Suc_times_binomial_eq
tff(fact_6592_binomial__symmetric,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)) ) ) ).

% binomial_symmetric
tff(fact_6593_choose__mult__lemma,axiom,
    ! [M2: nat,R: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),R)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K2))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K2)),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),R)),K2)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),R)),M2)) ).

% choose_mult_lemma
tff(fact_6594_binomial__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),R)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R))) ) ).

% binomial_le_pow
tff(fact_6595_zero__less__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2))) ) ).

% zero_less_binomial
tff(fact_6596_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_6597_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2))),aa_nat_nat(suc,K2)) ).

% binomial_Suc_Suc_eq_times
tff(fact_6598_choose__mult,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),M2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M2),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K2))) ) ) ) ).

% choose_mult
tff(fact_6599_binomial__absorb__comp,axiom,
    ! [N: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K2)) ).

% binomial_absorb_comp
tff(fact_6600_binomial__absorption,axiom,
    ! [K2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K2)) ).

% binomial_absorption
tff(fact_6601_sum__bounds__lt__plus1,axiom,
    ! [F2: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gm(F2)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Mm)) ).

% sum_bounds_lt_plus1
tff(fact_6602_sum__bounds__lt__plus1,axiom,
    ! [F2: fun_nat_real,Mm: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gn(F2)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_real(groups6591440286371151544t_real(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Mm)) ).

% sum_bounds_lt_plus1
tff(fact_6603_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,K2))),K2)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_6604_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(semiri681578069525770553at_rat,K2))),K2)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_6605_binomial__le__pow2,axiom,
    ! [N: nat,K2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% binomial_le_pow2
tff(fact_6606_choose__reduce__nat,axiom,
    ! [N: nat,K2: 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(ord_less_nat,zero_zero_nat),K2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K2)) ) ) ) ).

% choose_reduce_nat
tff(fact_6607_times__binomial__minus1__eq,axiom,
    ! [K2: nat,N: 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(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_6608_binomial__addition__formula,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(suc,K2))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K2)) ) ) ).

% binomial_addition_formula
tff(fact_6609_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_ok(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_6610_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_ol(N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_6611_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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_om,N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_6612_choose__odd__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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_on,N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) ) ) ).

% choose_odd_sum
tff(fact_6613_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_oo(N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_6614_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_op(N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_6615_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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_oq,N),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_6616_choose__even__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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_or,N)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) ) ) ).

% choose_even_sum
tff(fact_6617_card__lists__distinct__length__eq,axiom,
    ! [A4: set_list_nat,K2: nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),finite_card_list_nat(A4)))
       => ( finite7325466520557071688st_nat(collec5989764272469232197st_nat(aa_nat3401988846471794802t_bool(aTP_Lamm_os(A4),K2))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A4)),K2)),one_one_nat)),finite_card_list_nat(A4))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_6618_card__lists__distinct__length__eq,axiom,
    ! [A4: set_set_nat,K2: nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),finite_card_set_nat(A4)))
       => ( finite5631907774883551598et_nat(collect_list_set_nat(aa_nat948039967059030092t_bool(aTP_Lamm_ot(A4),K2))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A4)),K2)),one_one_nat)),finite_card_set_nat(A4))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_6619_card__lists__distinct__length__eq,axiom,
    ! [A4: set_complex,K2: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),finite_card_complex(A4)))
       => ( finite5120063068150530198omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_ou(A4),K2))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A4)),K2)),one_one_nat)),finite_card_complex(A4))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_6620_card__lists__distinct__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,K2: nat] :
      ( finite5795047828879050333T_VEBT(A4)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),finite7802652506058667612T_VEBT(A4)))
       => ( finite5915292604075114978T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_ov(A4),K2))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite7802652506058667612T_VEBT(A4)),K2)),one_one_nat)),finite7802652506058667612T_VEBT(A4))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_6621_card__lists__distinct__length__eq,axiom,
    ! [A4: set_bool,K2: nat] :
      ( finite_finite_bool(A4)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),finite_card_bool(A4)))
       => ( finite1699927739564962373t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_ow(A4),K2))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_bool(A4)),K2)),one_one_nat)),finite_card_bool(A4))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_6622_card__lists__distinct__length__eq,axiom,
    ! [A4: set_int,K2: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),finite_card_int(A4)))
       => ( finite_card_list_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_ox(A4),K2))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A4)),K2)),one_one_nat)),finite_card_int(A4))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_6623_card__lists__distinct__length__eq,axiom,
    ! [A4: set_nat,K2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),aa_set_nat_nat(finite_card_nat,A4)))
       => ( finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_oy(A4),K2))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,A4)),K2)),one_one_nat)),aa_set_nat_nat(finite_card_nat,A4))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_6624_gchoose__row__sum__weighted,axiom,
    ! [R: complex,M2: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_oz(R)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M2)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M2))),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(R),aa_nat_nat(suc,M2))) ).

% gchoose_row_sum_weighted
tff(fact_6625_gchoose__row__sum__weighted,axiom,
    ! [R: rat,M2: nat] : groups2906978787729119204at_rat(aTP_Lamm_pa(R),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M2))),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R),aa_nat_nat(suc,M2))) ).

% gchoose_row_sum_weighted
tff(fact_6626_gchoose__row__sum__weighted,axiom,
    ! [R: real,M2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pb,R)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,R),aa_nat_nat(suc,M2))) ).

% gchoose_row_sum_weighted
tff(fact_6627_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_6628_flip__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_nat_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(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_6629_flip__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2159334234014336723it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,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(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_6630_flip__bit__0,axiom,
    ! [A: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(zero_n356916108424825756nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_6631_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_6632_atMost__iff,axiom,
    ! [I2: real,K2: real] :
      ( pp(aa_set_real_bool(member_real(I2),set_ord_atMost_real(K2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I2),K2)) ) ).

% atMost_iff
tff(fact_6633_atMost__iff,axiom,
    ! [I2: set_nat,K2: set_nat] :
      ( member_set_nat(I2,set_or4236626031148496127et_nat(K2))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(I2),K2)) ) ).

% atMost_iff
tff(fact_6634_atMost__iff,axiom,
    ! [I2: rat,K2: rat] :
      ( member_rat(I2,set_ord_atMost_rat(K2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I2),K2)) ) ).

% atMost_iff
tff(fact_6635_atMost__iff,axiom,
    ! [I2: num,K2: num] :
      ( member_num(I2,set_ord_atMost_num(K2))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I2),K2)) ) ).

% atMost_iff
tff(fact_6636_atMost__iff,axiom,
    ! [I2: int,K2: int] :
      ( pp(aa_set_int_bool(member_int(I2),set_ord_atMost_int(K2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K2)) ) ).

% atMost_iff
tff(fact_6637_atMost__iff,axiom,
    ! [I2: nat,K2: nat] :
      ( pp(aa_set_nat_bool(member_nat(I2),aa_nat_set_nat(set_ord_atMost_nat,K2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),K2)) ) ).

% atMost_iff
tff(fact_6638_of__bool__less__eq__iff,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_n2052037380579107095ol_rat(P2)),zero_n2052037380579107095ol_rat(Q)))
    <=> ( pp(P2)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_6639_of__bool__less__eq__iff,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_n2687167440665602831ol_nat(P2)),zero_n2687167440665602831ol_nat(Q)))
    <=> ( pp(P2)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_6640_of__bool__less__eq__iff,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(zero_n2684676970156552555ol_int,P2)),aa_bool_int(zero_n2684676970156552555ol_int,Q)))
    <=> ( pp(P2)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_6641_of__bool__less__eq__iff,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_n356916108424825756nteger(P2)),zero_n356916108424825756nteger(Q)))
    <=> ( pp(P2)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_6642_of__bool__eq_I1_J,axiom,
    zero_n1201886186963655149omplex(fFalse) = zero_zero_complex ).

% of_bool_eq(1)
tff(fact_6643_of__bool__eq_I1_J,axiom,
    zero_n3304061248610475627l_real(fFalse) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_6644_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat(fFalse) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_6645_of__bool__eq_I1_J,axiom,
    zero_n2687167440665602831ol_nat(fFalse) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_6646_of__bool__eq_I1_J,axiom,
    aa_bool_int(zero_n2684676970156552555ol_int,fFalse) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_6647_of__bool__eq_I1_J,axiom,
    zero_n356916108424825756nteger(fFalse) = zero_z3403309356797280102nteger ).

% of_bool_eq(1)
tff(fact_6648_of__bool__eq__0__iff,axiom,
    ! [P2: bool] :
      ( ( zero_n1201886186963655149omplex(P2) = zero_zero_complex )
    <=> ~ pp(P2) ) ).

% of_bool_eq_0_iff
tff(fact_6649_of__bool__eq__0__iff,axiom,
    ! [P2: bool] :
      ( ( zero_n3304061248610475627l_real(P2) = zero_zero_real )
    <=> ~ pp(P2) ) ).

% of_bool_eq_0_iff
tff(fact_6650_of__bool__eq__0__iff,axiom,
    ! [P2: bool] :
      ( ( zero_n2052037380579107095ol_rat(P2) = zero_zero_rat )
    <=> ~ pp(P2) ) ).

% of_bool_eq_0_iff
tff(fact_6651_of__bool__eq__0__iff,axiom,
    ! [P2: bool] :
      ( ( zero_n2687167440665602831ol_nat(P2) = zero_zero_nat )
    <=> ~ pp(P2) ) ).

% of_bool_eq_0_iff
tff(fact_6652_of__bool__eq__0__iff,axiom,
    ! [P2: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P2) = zero_zero_int )
    <=> ~ pp(P2) ) ).

% of_bool_eq_0_iff
tff(fact_6653_of__bool__eq__0__iff,axiom,
    ! [P2: bool] :
      ( ( zero_n356916108424825756nteger(P2) = zero_z3403309356797280102nteger )
    <=> ~ pp(P2) ) ).

% of_bool_eq_0_iff
tff(fact_6654_of__bool__less__iff,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_n3304061248610475627l_real(P2)),zero_n3304061248610475627l_real(Q)))
    <=> ( ~ pp(P2)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_6655_of__bool__less__iff,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_n2052037380579107095ol_rat(P2)),zero_n2052037380579107095ol_rat(Q)))
    <=> ( ~ pp(P2)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_6656_of__bool__less__iff,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_n2687167440665602831ol_nat(P2)),zero_n2687167440665602831ol_nat(Q)))
    <=> ( ~ pp(P2)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_6657_of__bool__less__iff,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_bool_int(zero_n2684676970156552555ol_int,P2)),aa_bool_int(zero_n2684676970156552555ol_int,Q)))
    <=> ( ~ pp(P2)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_6658_of__bool__less__iff,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_n356916108424825756nteger(P2)),zero_n356916108424825756nteger(Q)))
    <=> ( ~ pp(P2)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_6659_of__bool__eq__1__iff,axiom,
    ! [P2: bool] :
      ( ( zero_n1201886186963655149omplex(P2) = one_one_complex )
    <=> pp(P2) ) ).

% of_bool_eq_1_iff
tff(fact_6660_of__bool__eq__1__iff,axiom,
    ! [P2: bool] :
      ( ( zero_n3304061248610475627l_real(P2) = one_one_real )
    <=> pp(P2) ) ).

% of_bool_eq_1_iff
tff(fact_6661_of__bool__eq__1__iff,axiom,
    ! [P2: bool] :
      ( ( zero_n2687167440665602831ol_nat(P2) = one_one_nat )
    <=> pp(P2) ) ).

% of_bool_eq_1_iff
tff(fact_6662_of__bool__eq__1__iff,axiom,
    ! [P2: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P2) = one_one_int )
    <=> pp(P2) ) ).

% of_bool_eq_1_iff
tff(fact_6663_of__bool__eq__1__iff,axiom,
    ! [P2: bool] :
      ( ( zero_n356916108424825756nteger(P2) = one_one_Code_integer )
    <=> pp(P2) ) ).

% of_bool_eq_1_iff
tff(fact_6664_of__bool__eq_I2_J,axiom,
    zero_n1201886186963655149omplex(fTrue) = one_one_complex ).

% of_bool_eq(2)
tff(fact_6665_of__bool__eq_I2_J,axiom,
    zero_n3304061248610475627l_real(fTrue) = one_one_real ).

% of_bool_eq(2)
tff(fact_6666_of__bool__eq_I2_J,axiom,
    zero_n2687167440665602831ol_nat(fTrue) = one_one_nat ).

% of_bool_eq(2)
tff(fact_6667_of__bool__eq_I2_J,axiom,
    aa_bool_int(zero_n2684676970156552555ol_int,fTrue) = one_one_int ).

% of_bool_eq(2)
tff(fact_6668_of__bool__eq_I2_J,axiom,
    zero_n356916108424825756nteger(fTrue) = one_one_Code_integer ).

% of_bool_eq(2)
tff(fact_6669_of__nat__of__bool,axiom,
    ! [P2: bool] : aa_nat_real(semiri5074537144036343181t_real,zero_n2687167440665602831ol_nat(P2)) = zero_n3304061248610475627l_real(P2) ).

% of_nat_of_bool
tff(fact_6670_of__nat__of__bool,axiom,
    ! [P2: bool] : aa_nat_nat(semiri1316708129612266289at_nat,zero_n2687167440665602831ol_nat(P2)) = zero_n2687167440665602831ol_nat(P2) ).

% of_nat_of_bool
tff(fact_6671_of__nat__of__bool,axiom,
    ! [P2: bool] : aa_nat_int(semiri1314217659103216013at_int,zero_n2687167440665602831ol_nat(P2)) = aa_bool_int(zero_n2684676970156552555ol_int,P2) ).

% of_nat_of_bool
tff(fact_6672_of__nat__of__bool,axiom,
    ! [P2: bool] : aa_nat_Code_integer(semiri4939895301339042750nteger,zero_n2687167440665602831ol_nat(P2)) = zero_n356916108424825756nteger(P2) ).

% of_nat_of_bool
tff(fact_6673_of__bool__or__iff,axiom,
    ! [P2: bool,Q: bool] : zero_n2687167440665602831ol_nat(aa_bool_bool(fdisj(P2),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_n2687167440665602831ol_nat(P2)),zero_n2687167440665602831ol_nat(Q)) ).

% of_bool_or_iff
tff(fact_6674_of__bool__or__iff,axiom,
    ! [P2: bool,Q: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fdisj(P2),Q)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_bool_int(zero_n2684676970156552555ol_int,P2)),aa_bool_int(zero_n2684676970156552555ol_int,Q)) ).

% of_bool_or_iff
tff(fact_6675_of__bool__or__iff,axiom,
    ! [P2: bool,Q: bool] : zero_n356916108424825756nteger(aa_bool_bool(fdisj(P2),Q)) = ord_max_Code_integer(zero_n356916108424825756nteger(P2),zero_n356916108424825756nteger(Q)) ).

% of_bool_or_iff
tff(fact_6676_finite__atMost,axiom,
    ! [K2: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_ord_atMost_nat,K2))) ).

% finite_atMost
tff(fact_6677_zero__less__of__bool__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),zero_n3304061248610475627l_real(P2)))
    <=> pp(P2) ) ).

% zero_less_of_bool_iff
tff(fact_6678_zero__less__of__bool__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),zero_n2052037380579107095ol_rat(P2)))
    <=> pp(P2) ) ).

% zero_less_of_bool_iff
tff(fact_6679_zero__less__of__bool__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),zero_n2687167440665602831ol_nat(P2)))
    <=> pp(P2) ) ).

% zero_less_of_bool_iff
tff(fact_6680_zero__less__of__bool__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_bool_int(zero_n2684676970156552555ol_int,P2)))
    <=> pp(P2) ) ).

% zero_less_of_bool_iff
tff(fact_6681_zero__less__of__bool__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger(P2)))
    <=> pp(P2) ) ).

% zero_less_of_bool_iff
tff(fact_6682_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_6683_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_6684_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_6685_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_6686_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_6687_of__bool__less__one__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_n3304061248610475627l_real(P2)),one_one_real))
    <=> ~ pp(P2) ) ).

% of_bool_less_one_iff
tff(fact_6688_of__bool__less__one__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_n2052037380579107095ol_rat(P2)),one_one_rat))
    <=> ~ pp(P2) ) ).

% of_bool_less_one_iff
tff(fact_6689_of__bool__less__one__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_n2687167440665602831ol_nat(P2)),one_one_nat))
    <=> ~ pp(P2) ) ).

% of_bool_less_one_iff
tff(fact_6690_of__bool__less__one__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_bool_int(zero_n2684676970156552555ol_int,P2)),one_one_int))
    <=> ~ pp(P2) ) ).

% of_bool_less_one_iff
tff(fact_6691_of__bool__less__one__iff,axiom,
    ! [P2: bool] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_n356916108424825756nteger(P2)),one_one_Code_integer))
    <=> ~ pp(P2) ) ).

% of_bool_less_one_iff
tff(fact_6692_of__bool__not__iff,axiom,
    ! [P2: bool] : zero_n1201886186963655149omplex(aa_bool_bool(fNot,P2)) = aa_complex_complex(minus_minus_complex(one_one_complex),zero_n1201886186963655149omplex(P2)) ).

% of_bool_not_iff
tff(fact_6693_of__bool__not__iff,axiom,
    ! [P2: bool] : zero_n3304061248610475627l_real(aa_bool_bool(fNot,P2)) = aa_real_real(minus_minus_real(one_one_real),zero_n3304061248610475627l_real(P2)) ).

% of_bool_not_iff
tff(fact_6694_of__bool__not__iff,axiom,
    ! [P2: bool] : zero_n2052037380579107095ol_rat(aa_bool_bool(fNot,P2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),zero_n2052037380579107095ol_rat(P2)) ).

% of_bool_not_iff
tff(fact_6695_of__bool__not__iff,axiom,
    ! [P2: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,P2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_bool_int(zero_n2684676970156552555ol_int,P2)) ).

% of_bool_not_iff
tff(fact_6696_of__bool__not__iff,axiom,
    ! [P2: bool] : zero_n356916108424825756nteger(aa_bool_bool(fNot,P2)) = minus_8373710615458151222nteger(one_one_Code_integer,zero_n356916108424825756nteger(P2)) ).

% of_bool_not_iff
tff(fact_6697_Suc__0__mod__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),N) = zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),aa_nat_nat(suc,zero_zero_nat)))) ).

% Suc_0_mod_eq
tff(fact_6698_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_complex(gbinomial_complex(zero_zero_complex),aa_nat_nat(suc,K2)) = zero_zero_complex ).

% gbinomial_0(2)
tff(fact_6699_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,zero_zero_real),aa_nat_nat(suc,K2)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_6700_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_rat(gbinomial_rat(zero_zero_rat),aa_nat_nat(suc,K2)) = zero_zero_rat ).

% gbinomial_0(2)
tff(fact_6701_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,zero_zero_nat),aa_nat_nat(suc,K2)) = zero_zero_nat ).

% gbinomial_0(2)
tff(fact_6702_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_int(gbinomial_int(zero_zero_int),aa_nat_nat(suc,K2)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_6703_gbinomial__0_I1_J,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(gbinom8545251970709558553nteger(A),zero_zero_nat) = one_one_Code_integer ).

% gbinomial_0(1)
tff(fact_6704_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_6705_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_6706_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_6707_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_6708_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_6709_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_6710_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_6711_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_6712_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_6713_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_6714_Icc__subset__Iic__iff,axiom,
    ! [L: real,H: real,H2: real] :
      ( pp(aa_set_real_bool(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_6715_sum_OatMost__Suc,axiom,
    ! [G2: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_rat(G2,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_6716_sum_OatMost__Suc,axiom,
    ! [G2: fun_nat_int,N: nat] : groups3539618377306564664at_int(G2,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G2,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_int(G2,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_6717_sum_OatMost__Suc,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_nat(G2,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_6718_sum_OatMost__Suc,axiom,
    ! [G2: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_real(G2,aa_nat_nat(suc,N))) ).

% sum.atMost_Suc
tff(fact_6719_prod_OatMost__Suc,axiom,
    ! [G2: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_real(G2,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_6720_prod_OatMost__Suc,axiom,
    ! [G2: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G2,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G2,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_rat(G2,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_6721_prod_OatMost__Suc,axiom,
    ! [G2: fun_nat_int,N: nat] : groups705719431365010083at_int(G2,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G2,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_int(G2,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_6722_prod_OatMost__Suc,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_nat(G2,aa_nat_nat(suc,N))) ).

% prod.atMost_Suc
tff(fact_6723_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_6724_distinct__swap,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_lis2822252094023462681T_bool(distinct_VEBT_VEBT,list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,nth_VEBT_VEBT(Xs,J2)),J2,nth_VEBT_VEBT(Xs,I2))))
        <=> pp(aa_lis2822252094023462681T_bool(distinct_VEBT_VEBT,Xs)) ) ) ) ).

% distinct_swap
tff(fact_6725_distinct__swap,axiom,
    ! [I2: nat,Xs: list_bool,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_bool(Xs)))
       => ( pp(aa_list_bool_bool(distinct_bool,list_update_bool(list_update_bool(Xs,I2,aa_nat_bool(nth_bool(Xs),J2)),J2,aa_nat_bool(nth_bool(Xs),I2))))
        <=> pp(aa_list_bool_bool(distinct_bool,Xs)) ) ) ) ).

% distinct_swap
tff(fact_6726_distinct__swap,axiom,
    ! [I2: nat,Xs: list_nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( pp(aa_list_nat_bool(distinct_nat,list_update_nat(list_update_nat(Xs,I2,aa_nat_nat(nth_nat(Xs),J2)),J2,aa_nat_nat(nth_nat(Xs),I2))))
        <=> pp(aa_list_nat_bool(distinct_nat,Xs)) ) ) ) ).

% distinct_swap
tff(fact_6727_distinct__swap,axiom,
    ! [I2: nat,Xs: list_int,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_int(Xs)))
       => ( pp(aa_list_int_bool(distinct_int,list_update_int(list_update_int(Xs,I2,aa_nat_int(nth_int(Xs),J2)),J2,aa_nat_int(nth_int(Xs),I2))))
        <=> pp(aa_list_int_bool(distinct_int,Xs)) ) ) ) ).

% distinct_swap
tff(fact_6728_sum__mult__of__bool__eq,axiom,
    ! [A4: set_real,F2: fun_real_real,P2: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups8097168146408367636l_real(aa_fun6348708079608492509l_real(aTP_Lamm_pc(F2),P2),A4) = groups8097168146408367636l_real(F2,inf_inf_set_real(A4,collect_real(P2))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_6729_sum__mult__of__bool__eq,axiom,
    ! [A4: set_int,F2: fun_int_real,P2: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups8778361861064173332t_real(aa_fun6974987029372589917t_real(aTP_Lamm_pd(F2),P2),A4) = groups8778361861064173332t_real(F2,inf_inf_set_int(A4,collect_int(P2))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_6730_sum__mult__of__bool__eq,axiom,
    ! [A4: set_complex,F2: fun_complex_real,P2: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5808333547571424918x_real(aa_fun2902006202505214429x_real(aTP_Lamm_pe(F2),P2),A4) = groups5808333547571424918x_real(F2,inf_inf_set_complex(A4,collect_complex(P2))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_6731_sum__mult__of__bool__eq,axiom,
    ! [A4: set_real,F2: fun_real_rat,P2: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1300246762558778688al_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_pf(F2),P2),A4) = groups1300246762558778688al_rat(F2,inf_inf_set_real(A4,collect_real(P2))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_6732_sum__mult__of__bool__eq,axiom,
    ! [A4: set_int,F2: fun_int_rat,P2: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups3906332499630173760nt_rat(aa_fun4400602662665189257nt_rat(aTP_Lamm_pg(F2),P2),A4) = groups3906332499630173760nt_rat(F2,inf_inf_set_int(A4,collect_int(P2))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_6733_sum__mult__of__bool__eq,axiom,
    ! [A4: set_complex,F2: fun_complex_rat,P2: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5058264527183730370ex_rat(aa_fun5529999419959482889ex_rat(aTP_Lamm_ph(F2),P2),A4) = groups5058264527183730370ex_rat(F2,inf_inf_set_complex(A4,collect_complex(P2))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_6734_sum__mult__of__bool__eq,axiom,
    ! [A4: set_nat,F2: fun_nat_rat,P2: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( groups2906978787729119204at_rat(aa_fun5656162108953708681at_rat(aTP_Lamm_pi(F2),P2),A4) = groups2906978787729119204at_rat(F2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(P2))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_6735_sum__mult__of__bool__eq,axiom,
    ! [A4: set_real,F2: fun_real_nat,P2: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1935376822645274424al_nat(aa_fun1504556115071737857al_nat(aTP_Lamm_pj(F2),P2),A4) = groups1935376822645274424al_nat(F2,inf_inf_set_real(A4,collect_real(P2))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_6736_sum__mult__of__bool__eq,axiom,
    ! [A4: set_int,F2: fun_int_nat,P2: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups4541462559716669496nt_nat(aa_fun7908831221058585985nt_nat(aTP_Lamm_pk(F2),P2),A4) = groups4541462559716669496nt_nat(F2,inf_inf_set_int(A4,collect_int(P2))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_6737_sum__mult__of__bool__eq,axiom,
    ! [A4: set_complex,F2: fun_complex_nat,P2: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5693394587270226106ex_nat(aa_fun9038227978352879617ex_nat(aTP_Lamm_pl(F2),P2),A4) = groups5693394587270226106ex_nat(F2,inf_inf_set_complex(A4,collect_complex(P2))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_6738_sum__of__bool__mult__eq,axiom,
    ! [A4: set_real,P2: fun_real_bool,F2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_pm(P2),F2),A4) = groups8097168146408367636l_real(F2,inf_inf_set_real(A4,collect_real(P2))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_6739_sum__of__bool__mult__eq,axiom,
    ! [A4: set_int,P2: fun_int_bool,F2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_pn(P2),F2),A4) = groups8778361861064173332t_real(F2,inf_inf_set_int(A4,collect_int(P2))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_6740_sum__of__bool__mult__eq,axiom,
    ! [A4: set_complex,P2: fun_complex_bool,F2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_po(P2),F2),A4) = groups5808333547571424918x_real(F2,inf_inf_set_complex(A4,collect_complex(P2))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_6741_sum__of__bool__mult__eq,axiom,
    ! [A4: set_real,P2: fun_real_bool,F2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_pp(P2),F2),A4) = groups1300246762558778688al_rat(F2,inf_inf_set_real(A4,collect_real(P2))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_6742_sum__of__bool__mult__eq,axiom,
    ! [A4: set_int,P2: fun_int_bool,F2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_pq(P2),F2),A4) = groups3906332499630173760nt_rat(F2,inf_inf_set_int(A4,collect_int(P2))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_6743_sum__of__bool__mult__eq,axiom,
    ! [A4: set_complex,P2: fun_complex_bool,F2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_pr(P2),F2),A4) = groups5058264527183730370ex_rat(F2,inf_inf_set_complex(A4,collect_complex(P2))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_6744_sum__of__bool__mult__eq,axiom,
    ! [A4: set_nat,P2: fun_nat_bool,F2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ps(P2),F2),A4) = groups2906978787729119204at_rat(F2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(P2))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_6745_sum__of__bool__mult__eq,axiom,
    ! [A4: set_real,P2: fun_real_bool,F2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_pt(P2),F2),A4) = groups1935376822645274424al_nat(F2,inf_inf_set_real(A4,collect_real(P2))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_6746_sum__of__bool__mult__eq,axiom,
    ! [A4: set_int,P2: fun_int_bool,F2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_pu(P2),F2),A4) = groups4541462559716669496nt_nat(F2,inf_inf_set_int(A4,collect_int(P2))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_6747_sum__of__bool__mult__eq,axiom,
    ! [A4: set_complex,P2: fun_complex_bool,F2: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_pv(P2),F2),A4) = groups5693394587270226106ex_nat(F2,inf_inf_set_complex(A4,collect_complex(P2))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_6748_of__bool__half__eq__0,axiom,
    ! [B: bool] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_n2687167440665602831ol_nat(B)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_6749_of__bool__half__eq__0,axiom,
    ! [B: bool] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_bool_int(zero_n2684676970156552555ol_int,B)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_6750_of__bool__half__eq__0,axiom,
    ! [B: bool] : divide6298287555418463151nteger(zero_n356916108424825756nteger(B),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_6751_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_ou(A4),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_6752_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A4)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_ov(A4),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_6753_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_bool,N: nat] :
      ( finite_finite_bool(A4)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_ow(A4),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_6754_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_ox(A4),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_6755_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_oy(A4),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_6756_one__div__2__pow__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_6757_one__div__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_6758_one__div__2__pow__eq,axiom,
    ! [N: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_6759_bits__1__div__exp,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_6760_bits__1__div__exp,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_6761_bits__1__div__exp,axiom,
    ! [N: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_6762_one__mod__2__pow__eq,axiom,
    ! [N: nat] : modulo8411746178871703098atural(one_one_Code_natural,power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),N)) = zero_n8403883297036319079atural(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_6763_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_6764_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_6765_one__mod__2__pow__eq,axiom,
    ! [N: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_6766_of__bool__eq__iff,axiom,
    ! [P: bool,Q2: bool] :
      ( ( zero_n2687167440665602831ol_nat(P) = zero_n2687167440665602831ol_nat(Q2) )
    <=> ( pp(P)
      <=> pp(Q2) ) ) ).

% of_bool_eq_iff
tff(fact_6767_of__bool__eq__iff,axiom,
    ! [P: bool,Q2: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P) = aa_bool_int(zero_n2684676970156552555ol_int,Q2) )
    <=> ( pp(P)
      <=> pp(Q2) ) ) ).

% of_bool_eq_iff
tff(fact_6768_of__bool__eq__iff,axiom,
    ! [P: bool,Q2: bool] :
      ( ( zero_n356916108424825756nteger(P) = zero_n356916108424825756nteger(Q2) )
    <=> ( pp(P)
      <=> pp(Q2) ) ) ).

% of_bool_eq_iff
tff(fact_6769_of__bool__conj,axiom,
    ! [P2: bool,Q: bool] : zero_n3304061248610475627l_real(aa_bool_bool(fconj(P2),Q)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_n3304061248610475627l_real(P2)),zero_n3304061248610475627l_real(Q)) ).

% of_bool_conj
tff(fact_6770_of__bool__conj,axiom,
    ! [P2: bool,Q: bool] : zero_n2052037380579107095ol_rat(aa_bool_bool(fconj(P2),Q)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_n2052037380579107095ol_rat(P2)),zero_n2052037380579107095ol_rat(Q)) ).

% of_bool_conj
tff(fact_6771_of__bool__conj,axiom,
    ! [P2: bool,Q: bool] : zero_n2687167440665602831ol_nat(aa_bool_bool(fconj(P2),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(P2)),zero_n2687167440665602831ol_nat(Q)) ).

% of_bool_conj
tff(fact_6772_of__bool__conj,axiom,
    ! [P2: bool,Q: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(P2),Q)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,P2)),aa_bool_int(zero_n2684676970156552555ol_int,Q)) ).

% of_bool_conj
tff(fact_6773_of__bool__conj,axiom,
    ! [P2: bool,Q: bool] : zero_n356916108424825756nteger(aa_bool_bool(fconj(P2),Q)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(P2)),zero_n356916108424825756nteger(Q)) ).

% of_bool_conj
tff(fact_6774_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_6775_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_6776_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_6777_infinite__Iic,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(finite_finite_int,set_ord_atMost_int(A))) ).

% infinite_Iic
tff(fact_6778_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_6779_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_6780_atMost__def,axiom,
    ! [U: real] : set_ord_atMost_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_pw,U)) ).

% atMost_def
tff(fact_6781_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aTP_Lamm_px(U)) ).

% atMost_def
tff(fact_6782_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_py,U)) ).

% atMost_def
tff(fact_6783_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_pz,U)) ).

% atMost_def
tff(fact_6784_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_qa,U)) ).

% atMost_def
tff(fact_6785_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ah,U)) ).

% atMost_def
tff(fact_6786_zero__less__eq__of__bool,axiom,
    ! [P2: bool] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),zero_n3304061248610475627l_real(P2))) ).

% zero_less_eq_of_bool
tff(fact_6787_zero__less__eq__of__bool,axiom,
    ! [P2: bool] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),zero_n2052037380579107095ol_rat(P2))) ).

% zero_less_eq_of_bool
tff(fact_6788_zero__less__eq__of__bool,axiom,
    ! [P2: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),zero_n2687167440665602831ol_nat(P2))) ).

% zero_less_eq_of_bool
tff(fact_6789_zero__less__eq__of__bool,axiom,
    ! [P2: bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_bool_int(zero_n2684676970156552555ol_int,P2))) ).

% zero_less_eq_of_bool
tff(fact_6790_zero__less__eq__of__bool,axiom,
    ! [P2: bool] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger(P2))) ).

% zero_less_eq_of_bool
tff(fact_6791_of__bool__less__eq__one,axiom,
    ! [P2: bool] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_n3304061248610475627l_real(P2)),one_one_real)) ).

% of_bool_less_eq_one
tff(fact_6792_of__bool__less__eq__one,axiom,
    ! [P2: bool] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_n2052037380579107095ol_rat(P2)),one_one_rat)) ).

% of_bool_less_eq_one
tff(fact_6793_of__bool__less__eq__one,axiom,
    ! [P2: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_n2687167440665602831ol_nat(P2)),one_one_nat)) ).

% of_bool_less_eq_one
tff(fact_6794_of__bool__less__eq__one,axiom,
    ! [P2: bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(zero_n2684676970156552555ol_int,P2)),one_one_int)) ).

% of_bool_less_eq_one
tff(fact_6795_of__bool__less__eq__one,axiom,
    ! [P2: bool] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_n356916108424825756nteger(P2)),one_one_Code_integer)) ).

% of_bool_less_eq_one
tff(fact_6796_of__bool__def,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( zero_n1201886186963655149omplex(P) = one_one_complex ) )
      & ( ~ pp(P)
       => ( zero_n1201886186963655149omplex(P) = zero_zero_complex ) ) ) ).

% of_bool_def
tff(fact_6797_of__bool__def,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( zero_n3304061248610475627l_real(P) = one_one_real ) )
      & ( ~ pp(P)
       => ( zero_n3304061248610475627l_real(P) = zero_zero_real ) ) ) ).

% of_bool_def
tff(fact_6798_of__bool__def,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( zero_n2052037380579107095ol_rat(P) = one_one_rat ) )
      & ( ~ pp(P)
       => ( zero_n2052037380579107095ol_rat(P) = zero_zero_rat ) ) ) ).

% of_bool_def
tff(fact_6799_of__bool__def,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( zero_n2687167440665602831ol_nat(P) = one_one_nat ) )
      & ( ~ pp(P)
       => ( zero_n2687167440665602831ol_nat(P) = zero_zero_nat ) ) ) ).

% of_bool_def
tff(fact_6800_of__bool__def,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( aa_bool_int(zero_n2684676970156552555ol_int,P) = one_one_int ) )
      & ( ~ pp(P)
       => ( aa_bool_int(zero_n2684676970156552555ol_int,P) = zero_zero_int ) ) ) ).

% of_bool_def
tff(fact_6801_of__bool__def,axiom,
    ! [P: bool] :
      ( ( pp(P)
       => ( zero_n356916108424825756nteger(P) = one_one_Code_integer ) )
      & ( ~ pp(P)
       => ( zero_n356916108424825756nteger(P) = zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
tff(fact_6802_split__of__bool,axiom,
    ! [P2: fun_complex_bool,P: bool] :
      ( pp(aa_complex_bool(P2,zero_n1201886186963655149omplex(P)))
    <=> ( ( pp(P)
         => pp(aa_complex_bool(P2,one_one_complex)) )
        & ( ~ pp(P)
         => pp(aa_complex_bool(P2,zero_zero_complex)) ) ) ) ).

% split_of_bool
tff(fact_6803_split__of__bool,axiom,
    ! [P2: fun_real_bool,P: bool] :
      ( pp(aa_real_bool(P2,zero_n3304061248610475627l_real(P)))
    <=> ( ( pp(P)
         => pp(aa_real_bool(P2,one_one_real)) )
        & ( ~ pp(P)
         => pp(aa_real_bool(P2,zero_zero_real)) ) ) ) ).

% split_of_bool
tff(fact_6804_split__of__bool,axiom,
    ! [P2: fun_rat_bool,P: bool] :
      ( pp(aa_rat_bool(P2,zero_n2052037380579107095ol_rat(P)))
    <=> ( ( pp(P)
         => pp(aa_rat_bool(P2,one_one_rat)) )
        & ( ~ pp(P)
         => pp(aa_rat_bool(P2,zero_zero_rat)) ) ) ) ).

% split_of_bool
tff(fact_6805_split__of__bool,axiom,
    ! [P2: fun_nat_bool,P: bool] :
      ( pp(aa_nat_bool(P2,zero_n2687167440665602831ol_nat(P)))
    <=> ( ( pp(P)
         => pp(aa_nat_bool(P2,one_one_nat)) )
        & ( ~ pp(P)
         => pp(aa_nat_bool(P2,zero_zero_nat)) ) ) ) ).

% split_of_bool
tff(fact_6806_split__of__bool,axiom,
    ! [P2: fun_int_bool,P: bool] :
      ( pp(aa_int_bool(P2,aa_bool_int(zero_n2684676970156552555ol_int,P)))
    <=> ( ( pp(P)
         => pp(aa_int_bool(P2,one_one_int)) )
        & ( ~ pp(P)
         => pp(aa_int_bool(P2,zero_zero_int)) ) ) ) ).

% split_of_bool
tff(fact_6807_split__of__bool,axiom,
    ! [P2: fun_Co8822979462182099027r_bool,P: bool] :
      ( pp(aa_Code_integer_bool(P2,zero_n356916108424825756nteger(P)))
    <=> ( ( pp(P)
         => pp(aa_Code_integer_bool(P2,one_one_Code_integer)) )
        & ( ~ pp(P)
         => pp(aa_Code_integer_bool(P2,zero_z3403309356797280102nteger)) ) ) ) ).

% split_of_bool
tff(fact_6808_split__of__bool__asm,axiom,
    ! [P2: fun_complex_bool,P: bool] :
      ( pp(aa_complex_bool(P2,zero_n1201886186963655149omplex(P)))
    <=> ~ ( ( pp(P)
            & ~ pp(aa_complex_bool(P2,one_one_complex)) )
          | ( ~ pp(P)
            & ~ pp(aa_complex_bool(P2,zero_zero_complex)) ) ) ) ).

% split_of_bool_asm
tff(fact_6809_split__of__bool__asm,axiom,
    ! [P2: fun_real_bool,P: bool] :
      ( pp(aa_real_bool(P2,zero_n3304061248610475627l_real(P)))
    <=> ~ ( ( pp(P)
            & ~ pp(aa_real_bool(P2,one_one_real)) )
          | ( ~ pp(P)
            & ~ pp(aa_real_bool(P2,zero_zero_real)) ) ) ) ).

% split_of_bool_asm
tff(fact_6810_split__of__bool__asm,axiom,
    ! [P2: fun_rat_bool,P: bool] :
      ( pp(aa_rat_bool(P2,zero_n2052037380579107095ol_rat(P)))
    <=> ~ ( ( pp(P)
            & ~ pp(aa_rat_bool(P2,one_one_rat)) )
          | ( ~ pp(P)
            & ~ pp(aa_rat_bool(P2,zero_zero_rat)) ) ) ) ).

% split_of_bool_asm
tff(fact_6811_split__of__bool__asm,axiom,
    ! [P2: fun_nat_bool,P: bool] :
      ( pp(aa_nat_bool(P2,zero_n2687167440665602831ol_nat(P)))
    <=> ~ ( ( pp(P)
            & ~ pp(aa_nat_bool(P2,one_one_nat)) )
          | ( ~ pp(P)
            & ~ pp(aa_nat_bool(P2,zero_zero_nat)) ) ) ) ).

% split_of_bool_asm
tff(fact_6812_split__of__bool__asm,axiom,
    ! [P2: fun_int_bool,P: bool] :
      ( pp(aa_int_bool(P2,aa_bool_int(zero_n2684676970156552555ol_int,P)))
    <=> ~ ( ( pp(P)
            & ~ pp(aa_int_bool(P2,one_one_int)) )
          | ( ~ pp(P)
            & ~ pp(aa_int_bool(P2,zero_zero_int)) ) ) ) ).

% split_of_bool_asm
tff(fact_6813_split__of__bool__asm,axiom,
    ! [P2: fun_Co8822979462182099027r_bool,P: bool] :
      ( pp(aa_Code_integer_bool(P2,zero_n356916108424825756nteger(P)))
    <=> ~ ( ( pp(P)
            & ~ pp(aa_Code_integer_bool(P2,one_one_Code_integer)) )
          | ( ~ pp(P)
            & ~ pp(aa_Code_integer_bool(P2,zero_z3403309356797280102nteger)) ) ) ) ).

% split_of_bool_asm
tff(fact_6814_atMost__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N) ).

% atMost_atLeast0
tff(fact_6815_lessThan__Suc__atMost,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K2)) = aa_nat_set_nat(set_ord_atMost_nat,K2) ).

% lessThan_Suc_atMost
tff(fact_6816_atMost__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K2)) = insert_nat(aa_nat_nat(suc,K2),aa_nat_set_nat(set_ord_atMost_nat,K2)) ).

% atMost_Suc
tff(fact_6817_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_6818_not__Iic__le__Icc,axiom,
    ! [H: real,L2: real,H2: real] : ~ pp(aa_set_real_bool(ord_less_eq_set_real(set_ord_atMost_real(H)),set_or1222579329274155063t_real(L2,H2))) ).

% not_Iic_le_Icc
tff(fact_6819_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_VEBT_VEBT,I2: nat,J2: nat] :
      ( pp(aa_lis2822252094023462681T_bool(distinct_VEBT_VEBT,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_s6755466524823107622T_VEBT(Xs)))
         => ( ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Xs,J2) )
          <=> ( I2 = J2 ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_6820_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_bool,I2: nat,J2: nat] :
      ( pp(aa_list_bool_bool(distinct_bool,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_bool(Xs)))
         => ( ( pp(aa_nat_bool(nth_bool(Xs),I2))
            <=> pp(aa_nat_bool(nth_bool(Xs),J2)) )
          <=> ( I2 = J2 ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_6821_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_nat,I2: nat,J2: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),aa_list_nat_nat(size_size_list_nat,Xs)))
         => ( ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Xs),J2) )
          <=> ( I2 = J2 ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_6822_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_int,I2: nat,J2: nat] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_int(Xs)))
         => ( ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Xs),J2) )
          <=> ( I2 = J2 ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_6823_distinct__conv__nth,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(distinct_VEBT_VEBT,Xs))
    <=> ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
         => ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs)))
             => ( ( I != J )
               => ( nth_VEBT_VEBT(Xs,I) != nth_VEBT_VEBT(Xs,J) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_6824_distinct__conv__nth,axiom,
    ! [Xs: list_bool] :
      ( pp(aa_list_bool_bool(distinct_bool,Xs))
    <=> ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
         => ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_bool(Xs)))
             => ( ( I != J )
               => ~ ( pp(aa_nat_bool(nth_bool(Xs),I))
                  <=> pp(aa_nat_bool(nth_bool(Xs),J)) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_6825_distinct__conv__nth,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
    <=> ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
         => ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,Xs)))
             => ( ( I != J )
               => ( aa_nat_nat(nth_nat(Xs),I) != aa_nat_nat(nth_nat(Xs),J) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_6826_distinct__conv__nth,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
    <=> ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
         => ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_int(Xs)))
             => ( ( I != J )
               => ( aa_nat_int(nth_int(Xs),I) != aa_nat_int(nth_int(Xs),J) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_6827_card__distinct,axiom,
    ! [Xs: list_complex] :
      ( ( finite_card_complex(set_complex2(Xs)) = size_s3451745648224563538omplex(Xs) )
     => distinct_complex(Xs) ) ).

% card_distinct
tff(fact_6828_card__distinct,axiom,
    ! [Xs: list_list_nat] :
      ( ( finite_card_list_nat(set_list_nat2(Xs)) = size_s3023201423986296836st_nat(Xs) )
     => distinct_list_nat(Xs) ) ).

% card_distinct
tff(fact_6829_card__distinct,axiom,
    ! [Xs: list_set_nat] :
      ( ( finite_card_set_nat(set_set_nat2(Xs)) = size_s3254054031482475050et_nat(Xs) )
     => distinct_set_nat(Xs) ) ).

% card_distinct
tff(fact_6830_card__distinct,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( ( finite7802652506058667612T_VEBT(set_VEBT_VEBT2(Xs)) = size_s6755466524823107622T_VEBT(Xs) )
     => pp(aa_lis2822252094023462681T_bool(distinct_VEBT_VEBT,Xs)) ) ).

% card_distinct
tff(fact_6831_card__distinct,axiom,
    ! [Xs: list_bool] :
      ( ( finite_card_bool(set_bool2(Xs)) = size_size_list_bool(Xs) )
     => pp(aa_list_bool_bool(distinct_bool,Xs)) ) ).

% card_distinct
tff(fact_6832_card__distinct,axiom,
    ! [Xs: list_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,set_nat2(Xs)) = aa_list_nat_nat(size_size_list_nat,Xs) )
     => pp(aa_list_nat_bool(distinct_nat,Xs)) ) ).

% card_distinct
tff(fact_6833_card__distinct,axiom,
    ! [Xs: list_int] :
      ( ( finite_card_int(set_int2(Xs)) = size_size_list_int(Xs) )
     => pp(aa_list_int_bool(distinct_int,Xs)) ) ).

% card_distinct
tff(fact_6834_distinct__card,axiom,
    ! [Xs: list_complex] :
      ( distinct_complex(Xs)
     => ( finite_card_complex(set_complex2(Xs)) = size_s3451745648224563538omplex(Xs) ) ) ).

% distinct_card
tff(fact_6835_distinct__card,axiom,
    ! [Xs: list_list_nat] :
      ( distinct_list_nat(Xs)
     => ( finite_card_list_nat(set_list_nat2(Xs)) = size_s3023201423986296836st_nat(Xs) ) ) ).

% distinct_card
tff(fact_6836_distinct__card,axiom,
    ! [Xs: list_set_nat] :
      ( distinct_set_nat(Xs)
     => ( finite_card_set_nat(set_set_nat2(Xs)) = size_s3254054031482475050et_nat(Xs) ) ) ).

% distinct_card
tff(fact_6837_distinct__card,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(distinct_VEBT_VEBT,Xs))
     => ( finite7802652506058667612T_VEBT(set_VEBT_VEBT2(Xs)) = size_s6755466524823107622T_VEBT(Xs) ) ) ).

% distinct_card
tff(fact_6838_distinct__card,axiom,
    ! [Xs: list_bool] :
      ( pp(aa_list_bool_bool(distinct_bool,Xs))
     => ( finite_card_bool(set_bool2(Xs)) = size_size_list_bool(Xs) ) ) ).

% distinct_card
tff(fact_6839_distinct__card,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( aa_set_nat_nat(finite_card_nat,set_nat2(Xs)) = aa_list_nat_nat(size_size_list_nat,Xs) ) ) ).

% distinct_card
tff(fact_6840_distinct__card,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( finite_card_int(set_int2(Xs)) = size_size_list_int(Xs) ) ) ).

% distinct_card
tff(fact_6841_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(A),K2)),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2))) ).

% gbinomial_Suc_Suc
tff(fact_6842_gbinomial__Suc__Suc,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K2)) = aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2))) ).

% gbinomial_Suc_Suc
tff(fact_6843_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(A),K2)),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2))) ).

% gbinomial_Suc_Suc
tff(fact_6844_gbinomial__of__nat__symmetric,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,N)),K2) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_6845_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_set_real_bool(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_6846_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_6847_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_6848_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_6849_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_6850_sum__choose__upper,axiom,
    ! [M2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qb,M2)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,M2)) ).

% sum_choose_upper
tff(fact_6851_distinct__Ex1,axiom,
    ! [Xs: list_complex,X: complex] :
      ( distinct_complex(Xs)
     => ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
       => ? [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),size_s3451745648224563538omplex(Xs)))
            & ( aa_nat_complex(nth_complex(Xs),X3) = X )
            & ! [Y4: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),size_s3451745648224563538omplex(Xs)))
                  & ( aa_nat_complex(nth_complex(Xs),Y4) = X ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_6852_distinct__Ex1,axiom,
    ! [Xs: list_real,X: real] :
      ( distinct_real(Xs)
     => ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
       => ? [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),size_size_list_real(Xs)))
            & ( aa_nat_real(nth_real(Xs),X3) = X )
            & ! [Y4: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),size_size_list_real(Xs)))
                  & ( aa_nat_real(nth_real(Xs),Y4) = X ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_6853_distinct__Ex1,axiom,
    ! [Xs: list_set_nat,X: set_nat] :
      ( distinct_set_nat(Xs)
     => ( member_set_nat(X,set_set_nat2(Xs))
       => ? [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),size_s3254054031482475050et_nat(Xs)))
            & ( aa_nat_set_nat(nth_set_nat(Xs),X3) = X )
            & ! [Y4: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),size_s3254054031482475050et_nat(Xs)))
                  & ( aa_nat_set_nat(nth_set_nat(Xs),Y4) = X ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_6854_distinct__Ex1,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(distinct_VEBT_VEBT,Xs))
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
       => ? [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),size_s6755466524823107622T_VEBT(Xs)))
            & ( nth_VEBT_VEBT(Xs,X3) = X )
            & ! [Y4: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),size_s6755466524823107622T_VEBT(Xs)))
                  & ( nth_VEBT_VEBT(Xs,Y4) = X ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_6855_distinct__Ex1,axiom,
    ! [Xs: list_bool,X: bool] :
      ( pp(aa_list_bool_bool(distinct_bool,Xs))
     => ( member_bool(X,set_bool2(Xs))
       => ? [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),size_size_list_bool(Xs)))
            & ( pp(aa_nat_bool(nth_bool(Xs),X3))
            <=> pp(X) )
            & ! [Y4: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),size_size_list_bool(Xs)))
                  & ( pp(aa_nat_bool(nth_bool(Xs),Y4))
                  <=> pp(X) ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_6856_distinct__Ex1,axiom,
    ! [Xs: list_nat,X: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
       => ? [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_list_nat_nat(size_size_list_nat,Xs)))
            & ( aa_nat_nat(nth_nat(Xs),X3) = X )
            & ! [Y4: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),aa_list_nat_nat(size_size_list_nat,Xs)))
                  & ( aa_nat_nat(nth_nat(Xs),Y4) = X ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_6857_distinct__Ex1,axiom,
    ! [Xs: list_int,X: int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
       => ? [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),size_size_list_int(Xs)))
            & ( aa_nat_int(nth_int(Xs),X3) = X )
            & ! [Y4: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),size_size_list_int(Xs)))
                  & ( aa_nat_int(nth_int(Xs),Y4) = X ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_6858_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M2: nat,A: complex,X: complex,Y: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_qc(M2),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_qd(M2),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M2)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_6859_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M2: nat,A: rat,X: rat,Y: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_qe(M2),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M2)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_qf(M2),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M2)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_6860_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M2: 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_qg(M2),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_qh(M2),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M2)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_6861_gbinomial__addition__formula,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(suc,K2))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ).

% gbinomial_addition_formula
tff(fact_6862_gbinomial__addition__formula,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2)) = aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),K2)) ).

% gbinomial_addition_formula
tff(fact_6863_gbinomial__addition__formula,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),K2)) ).

% gbinomial_addition_formula
tff(fact_6864_gbinomial__absorb__comp,axiom,
    ! [A: complex,K2: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),aa_nat_complex(gbinomial_complex(A),K2)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ).

% gbinomial_absorb_comp
tff(fact_6865_gbinomial__absorb__comp,axiom,
    ! [A: rat,K2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_rat(gbinomial_rat(A),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),K2)) ).

% gbinomial_absorb_comp
tff(fact_6866_gbinomial__absorb__comp,axiom,
    ! [A: real,K2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),K2)) ).

% gbinomial_absorb_comp
tff(fact_6867_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K2)),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(A),aa_nat_real(semiri5074537144036343181t_real,K2))),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_6868_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K2))),K2)),aa_nat_rat(gbinomial_rat(A),K2))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_6869_gbinomial__mult__1_H,axiom,
    ! [A: rat,K2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K2)),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)),aa_nat_rat(gbinomial_rat(A),K2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1'
tff(fact_6870_gbinomial__mult__1_H,axiom,
    ! [A: real,K2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)),A) = aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1'
tff(fact_6871_gbinomial__mult__1,axiom,
    ! [A: rat,K2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(A),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)),aa_nat_rat(gbinomial_rat(A),K2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1
tff(fact_6872_gbinomial__mult__1,axiom,
    ! [A: real,K2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) = aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1
tff(fact_6873_sum_OatMost__Suc__shift,axiom,
    ! [G2: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G2,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_gw(G2),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_6874_sum_OatMost__Suc__shift,axiom,
    ! [G2: fun_nat_int,N: nat] : groups3539618377306564664at_int(G2,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G2,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_gx(G2),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_6875_sum_OatMost__Suc__shift,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G2,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gm(G2)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_6876_sum_OatMost__Suc__shift,axiom,
    ! [G2: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(plus_plus_real(aa_nat_real(G2,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gn(G2)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% sum.atMost_Suc_shift
tff(fact_6877_sum__telescope,axiom,
    ! [F2: fun_nat_rat,I2: nat] : groups2906978787729119204at_rat(aTP_Lamm_lv(F2),aa_nat_set_nat(set_ord_atMost_nat,I2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(F2,zero_zero_nat)),aa_nat_rat(F2,aa_nat_nat(suc,I2))) ).

% sum_telescope
tff(fact_6878_sum__telescope,axiom,
    ! [F2: fun_nat_int,I2: nat] : groups3539618377306564664at_int(aTP_Lamm_lw(F2),aa_nat_set_nat(set_ord_atMost_nat,I2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F2,zero_zero_nat)),aa_nat_int(F2,aa_nat_nat(suc,I2))) ).

% sum_telescope
tff(fact_6879_sum__telescope,axiom,
    ! [F2: fun_nat_real,I2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lx(F2)),aa_nat_set_nat(set_ord_atMost_nat,I2)) = aa_real_real(minus_minus_real(aa_nat_real(F2,zero_zero_nat)),aa_nat_real(F2,aa_nat_nat(suc,I2))) ).

% sum_telescope
tff(fact_6880_polyfun__eq__coeffs,axiom,
    ! [C2: fun_nat_complex,N: nat,D: fun_nat_complex] :
      ( ! [X4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(C2),X4)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(D),X4)),aa_nat_set_nat(set_ord_atMost_nat,N))
    <=> ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
         => ( aa_nat_complex(C2,I) = aa_nat_complex(D,I) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_6881_polyfun__eq__coeffs,axiom,
    ! [C2: fun_nat_real,N: nat,D: fun_nat_real] :
      ( ! [X4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C2),X4)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(D),X4)),aa_nat_set_nat(set_ord_atMost_nat,N))
    <=> ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
         => ( aa_nat_real(C2,I) = aa_nat_real(D,I) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_6882_prod_OatMost__Suc__shift,axiom,
    ! [G2: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G2,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_gn(G2)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_6883_prod_OatMost__Suc__shift,axiom,
    ! [G2: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G2,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G2,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_gw(G2),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_6884_prod_OatMost__Suc__shift,axiom,
    ! [G2: fun_nat_int,N: nat] : groups705719431365010083at_int(G2,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G2,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_gx(G2),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_6885_prod_OatMost__Suc__shift,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G2,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_gm(G2)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% prod.atMost_Suc_shift
tff(fact_6886_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_qk(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qm(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nested_swap'
tff(fact_6887_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_qn(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_qp(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nested_swap'
tff(fact_6888_ivl__disj__un__one_I4_J,axiom,
    ! [L: rat,U: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),U))
     => ( sup_sup_set_rat(set_ord_lessThan_rat(L),set_or633870826150836451st_rat(L,U)) = set_ord_atMost_rat(U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_6889_ivl__disj__un__one_I4_J,axiom,
    ! [L: num,U: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),U))
     => ( sup_sup_set_num(set_ord_lessThan_num(L),set_or7049704709247886629st_num(L,U)) = set_ord_atMost_num(U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_6890_ivl__disj__un__one_I4_J,axiom,
    ! [L: nat,U: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),U))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,L)),aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = aa_nat_set_nat(set_ord_atMost_nat,U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_6891_ivl__disj__un__one_I4_J,axiom,
    ! [L: int,U: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),U))
     => ( sup_sup_set_int(set_ord_lessThan_int(L),set_or1266510415728281911st_int(L,U)) = set_ord_atMost_int(U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_6892_ivl__disj__un__one_I4_J,axiom,
    ! [L: real,U: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),U))
     => ( sup_sup_set_real(set_or5984915006950818249n_real(L),set_or1222579329274155063t_real(L,U)) = set_ord_atMost_real(U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_6893_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: int] : sup_sup_set_int(set_ord_lessThan_int(U),insert_int(U,bot_bot_set_int)) = set_ord_atMost_int(U) ).

% ivl_disj_un_singleton(2)
tff(fact_6894_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: real] : sup_sup_set_real(set_or5984915006950818249n_real(U),insert_real(U,bot_bot_set_real)) = set_ord_atMost_real(U) ).

% ivl_disj_un_singleton(2)
tff(fact_6895_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,U)),insert_nat(U,bot_bot_set_nat)) = aa_nat_set_nat(set_ord_atMost_nat,U) ).

% ivl_disj_un_singleton(2)
tff(fact_6896_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_int,N: nat] : groups705719431365010083at_int(aTP_Lamm_qq(A),aa_nat_set_nat(set_ord_atMost_nat,N)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qs(A),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nested_swap'
tff(fact_6897_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_qt(A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qu(A),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% prod.nested_swap'
tff(fact_6898_sum__choose__lower,axiom,
    ! [R: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qv,R)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R),N))),N) ).

% sum_choose_lower
tff(fact_6899_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2))),aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(gbinomial_complex(A),K2)) ).

% Suc_times_gbinomial
tff(fact_6900_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K2)) ).

% Suc_times_gbinomial
tff(fact_6901_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) ).

% Suc_times_gbinomial
tff(fact_6902_gbinomial__absorption,axiom,
    ! [K2: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ).

% gbinomial_absorption
tff(fact_6903_gbinomial__absorption,axiom,
    ! [K2: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),K2)) ).

% gbinomial_absorption
tff(fact_6904_gbinomial__absorption,axiom,
    ! [K2: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),K2)) ).

% gbinomial_absorption
tff(fact_6905_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,M2: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M2))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),M2)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,M2)),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K2))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_6906_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,M2: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),M2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,M2)),K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K2))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_6907_zero__polynom__imp__zero__coeffs,axiom,
    ! [C2: fun_nat_complex,N: nat,K2: nat] :
      ( ! [W: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(C2),W)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => ( aa_nat_complex(C2,K2) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_6908_zero__polynom__imp__zero__coeffs,axiom,
    ! [C2: fun_nat_real,N: nat,K2: nat] :
      ( ! [W: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C2),W)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => ( aa_nat_real(C2,K2) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_6909_polyfun__eq__0,axiom,
    ! [C2: fun_nat_complex,N: nat] :
      ( ! [X4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(C2),X4)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex
    <=> ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
         => ( aa_nat_complex(C2,I) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_6910_polyfun__eq__0,axiom,
    ! [C2: fun_nat_real,N: nat] :
      ( ! [X4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C2),X4)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real
    <=> ! [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
         => ( aa_nat_real(C2,I) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_6911_sum_OatMost__shift,axiom,
    ! [G2: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G2,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G2,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_gw(G2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_6912_sum_OatMost__shift,axiom,
    ! [G2: fun_nat_int,N: nat] : groups3539618377306564664at_int(G2,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G2,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_gx(G2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_6913_sum_OatMost__shift,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G2),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G2,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gm(G2)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_6914_sum_OatMost__shift,axiom,
    ! [G2: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G2),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(plus_plus_real(aa_nat_real(G2,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_gn(G2)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.atMost_shift
tff(fact_6915_sum__up__index__split,axiom,
    ! [F2: fun_nat_rat,M2: nat,N: nat] : groups2906978787729119204at_rat(F2,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(F2,aa_nat_set_nat(set_ord_atMost_nat,M2))),groups2906978787729119204at_rat(F2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)))) ).

% sum_up_index_split
tff(fact_6916_sum__up__index__split,axiom,
    ! [F2: fun_nat_int,M2: nat,N: nat] : groups3539618377306564664at_int(F2,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(F2,aa_nat_set_nat(set_ord_atMost_nat,M2))),groups3539618377306564664at_int(F2,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)))) ).

% sum_up_index_split
tff(fact_6917_sum__up__index__split,axiom,
    ! [F2: fun_nat_nat,M2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F2),aa_nat_set_nat(set_ord_atMost_nat,M2))),aa_set_nat_nat(groups3542108847815614940at_nat(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)))) ).

% sum_up_index_split
tff(fact_6918_sum__up__index__split,axiom,
    ! [F2: fun_nat_real,M2: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(F2),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(F2),aa_nat_set_nat(set_ord_atMost_nat,M2))),aa_set_nat_real(groups6591440286371151544t_real(F2),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)))) ).

% sum_up_index_split
tff(fact_6919_prod_OatMost__shift,axiom,
    ! [G2: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G2),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G2,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_gn(G2)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_6920_prod_OatMost__shift,axiom,
    ! [G2: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G2,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G2,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_gw(G2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_6921_prod_OatMost__shift,axiom,
    ! [G2: fun_nat_int,N: nat] : groups705719431365010083at_int(G2,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G2,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_gx(G2),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_6922_prod_OatMost__shift,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G2),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G2,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_gm(G2)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% prod.atMost_shift
tff(fact_6923_gbinomial__r__part__sum,axiom,
    ! [M2: nat] : groups2906978787729119204at_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,M2))),one_one_rat)),aa_nat_set_nat(set_ord_atMost_nat,M2)) = 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))),M2)) ).

% gbinomial_r_part_sum
tff(fact_6924_gbinomial__r__part__sum,axiom,
    ! [M2: nat] : aa_set_nat_complex(groups2073611262835488442omplex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,M2))),one_one_complex))),aa_nat_set_nat(set_ord_atMost_nat,M2)) = 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))),M2)) ).

% gbinomial_r_part_sum
tff(fact_6925_gbinomial__r__part__sum,axiom,
    ! [M2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(gbinomial_real,aa_real_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,M2))),one_one_real))),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)) ).

% gbinomial_r_part_sum
tff(fact_6926_sum__choose__diagonal,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qw(M2),N)),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),M2) ) ) ).

% sum_choose_diagonal
tff(fact_6927_vandermonde,axiom,
    ! [M2: nat,N: nat,R: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qx(M2),N),R)),aa_nat_set_nat(set_ord_atMost_nat,R)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)),R) ).

% vandermonde
tff(fact_6928_bits__induct,axiom,
    ! [P2: fun_nat_bool,A: nat] :
      ( ! [A3: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
         => pp(aa_nat_bool(P2,A3)) )
     => ( ! [A3: nat,B3: bool] :
            ( pp(aa_nat_bool(P2,A3))
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(B3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
             => pp(aa_nat_bool(P2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(B3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3)))) ) )
       => pp(aa_nat_bool(P2,A)) ) ) ).

% bits_induct
tff(fact_6929_bits__induct,axiom,
    ! [P2: fun_int_bool,A: int] :
      ( ! [A3: int] :
          ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
         => pp(aa_int_bool(P2,A3)) )
     => ( ! [A3: int,B3: bool] :
            ( pp(aa_int_bool(P2,A3))
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,B3)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
             => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,B3)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3)))) ) )
       => pp(aa_int_bool(P2,A)) ) ) ).

% bits_induct
tff(fact_6930_bits__induct,axiom,
    ! [P2: fun_Co8822979462182099027r_bool,A: code_integer] :
      ( ! [A3: code_integer] :
          ( ( divide6298287555418463151nteger(A3,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A3 )
         => pp(aa_Code_integer_bool(P2,A3)) )
     => ( ! [A3: code_integer,B3: bool] :
            ( pp(aa_Code_integer_bool(P2,A3))
           => ( ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(zero_n356916108424825756nteger(B3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A3)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = A3 )
             => pp(aa_Code_integer_bool(P2,plus_p5714425477246183910nteger(zero_n356916108424825756nteger(B3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A3)))) ) )
       => pp(aa_Code_integer_bool(P2,A)) ) ) ).

% bits_induct
tff(fact_6931_gbinomial__partial__row__sum,axiom,
    ! [A: complex,M2: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_oz(A)),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,M2)),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,M2),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_6932_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M2: nat] : groups2906978787729119204at_rat(aTP_Lamm_pa(A),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),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,M2),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_6933_gbinomial__partial__row__sum,axiom,
    ! [A: real,M2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pb,A)),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,M2)),one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_6934_gbinomial__factors,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2)))),aa_nat_complex(gbinomial_complex(A),K2)) ).

% gbinomial_factors
tff(fact_6935_gbinomial__factors,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2)))),aa_nat_rat(gbinomial_rat(A),K2)) ).

% gbinomial_factors
tff(fact_6936_gbinomial__factors,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) ).

% gbinomial_factors
tff(fact_6937_gbinomial__rec,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K2)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2)))) ).

% gbinomial_rec
tff(fact_6938_gbinomial__rec,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2)))) ).

% gbinomial_rec
tff(fact_6939_gbinomial__rec,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2)))) ).

% gbinomial_rec
tff(fact_6940_sum__gp__basic,axiom,
    ! [X: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(one_one_Code_integer,X)),groups7501900531339628137nteger(power_8256067586552552935nteger(X),aa_nat_set_nat(set_ord_atMost_nat,N))) = minus_8373710615458151222nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_6941_sum__gp__basic,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_6942_sum__gp__basic,axiom,
    ! [X: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_6943_sum__gp__basic,axiom,
    ! [X: int,N: 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,N))) = 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,N))) ).

% sum_gp_basic
tff(fact_6944_sum__gp__basic,axiom,
    ! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))) ).

% sum_gp_basic
tff(fact_6945_polyfun__finite__roots,axiom,
    ! [C2: fun_nat_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_qy(C2),N))))
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
          & ( aa_nat_complex(C2,I) != zero_zero_complex ) ) ) ).

% polyfun_finite_roots
tff(fact_6946_polyfun__finite__roots,axiom,
    ! [C2: fun_nat_real,N: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_qz(C2),N))))
    <=> ? [I: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
          & ( aa_nat_real(C2,I) != zero_zero_real ) ) ) ).

% polyfun_finite_roots
tff(fact_6947_polyfun__roots__finite,axiom,
    ! [C2: fun_nat_complex,K2: nat,N: nat] :
      ( ( aa_nat_complex(C2,K2) != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_qy(C2),N)))) ) ) ).

% polyfun_roots_finite
tff(fact_6948_polyfun__roots__finite,axiom,
    ! [C2: fun_nat_real,K2: nat,N: nat] :
      ( ( aa_nat_real(C2,K2) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_qz(C2),N)))) ) ) ).

% polyfun_roots_finite
tff(fact_6949_polyfun__roots__card,axiom,
    ! [C2: fun_nat_complex,K2: nat,N: nat] :
      ( ( aa_nat_complex(C2,K2) != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_qy(C2),N)))),N)) ) ) ).

% polyfun_roots_card
tff(fact_6950_polyfun__roots__card,axiom,
    ! [C2: fun_nat_real,K2: nat,N: nat] :
      ( ( aa_nat_real(C2,K2) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(collect_real(aa_nat_fun_real_bool(aTP_Lamm_qz(C2),N)))),N)) ) ) ).

% polyfun_roots_card
tff(fact_6951_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_complex,A: complex,N: nat] :
      ( ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(C2),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex )
     => ~ ! [B3: fun_nat_complex] :
            ~ ! [Z4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(C2),Z4)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z4),A)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(B3),Z4)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_6952_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_rat,A: rat,N: nat] :
      ( ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_ra(C2),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat )
     => ~ ! [B3: fun_nat_rat] :
            ~ ! [Z4: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_ra(C2),Z4),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Z4),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_ra(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_6953_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_int,A: int,N: nat] :
      ( ( groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rb(C2),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int )
     => ~ ! [B3: fun_nat_int] :
            ~ ! [Z4: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rb(C2),Z4),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z4),A)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rb(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_6954_polyfun__linear__factor__root,axiom,
    ! [C2: fun_nat_real,A: real,N: nat] :
      ( ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C2),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real )
     => ~ ! [B3: fun_nat_real] :
            ~ ! [Z4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C2),Z4)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(Z4),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(B3),Z4)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ).

% polyfun_linear_factor_root
tff(fact_6955_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_complex,N: nat,A: complex] :
    ? [B3: fun_nat_complex] :
    ! [Z4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(C2),Z4)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z4),A)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(B3),Z4)),aa_nat_set_nat(set_ord_lessThan_nat,N)))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(C2),A)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_6956_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_rat,N: nat,A: rat] :
    ? [B3: fun_nat_rat] :
    ! [Z4: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_ra(C2),Z4),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Z4),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_ra(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,N)))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_ra(C2),A),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_6957_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_int,N: nat,A: int] :
    ? [B3: fun_nat_int] :
    ! [Z4: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rb(C2),Z4),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z4),A)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rb(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,N)))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rb(C2),A),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_6958_polyfun__linear__factor,axiom,
    ! [C2: fun_nat_real,N: nat,A: real] :
    ? [B3: fun_nat_real] :
    ! [Z4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C2),Z4)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(Z4),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(B3),Z4)),aa_nat_set_nat(set_ord_lessThan_nat,N)))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C2),A)),aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% polyfun_linear_factor
tff(fact_6959_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M2)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ).

% sum_power_shift
tff(fact_6960_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),M2)),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,N),M2)))) ) ) ).

% sum_power_shift
tff(fact_6961_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),M2)),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,N),M2)))) ) ) ).

% sum_power_shift
tff(fact_6962_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M2)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ).

% sum_power_shift
tff(fact_6963_binomial,axiom,
    ! [A: nat,B: nat,N: 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)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_rc(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial
tff(fact_6964_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_atMost_nat,N)),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_6965_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] : modulo8411746178871703098atural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),M2),power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,zero_n8403883297036319079atural(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))),power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one)),M2)) ).

% exp_mod_exp
tff(fact_6966_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)) ).

% exp_mod_exp
tff(fact_6967_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)) ).

% exp_mod_exp
tff(fact_6968_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M2)) ).

% exp_mod_exp
tff(fact_6969_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
     => ( aa_nat_complex(gbinomial_complex(A),K2) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ) ) ).

% gbinomial_reduce_nat
tff(fact_6970_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),K2)) ) ) ).

% gbinomial_reduce_nat
tff(fact_6971_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
     => ( aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),K2)) ) ) ).

% gbinomial_reduce_nat
tff(fact_6972_sum_Oin__pairs__0,axiom,
    ! [G2: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G2,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))),N)))) = groups2906978787729119204at_rat(aTP_Lamm_hi(G2),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_6973_sum_Oin__pairs__0,axiom,
    ! [G2: fun_nat_int,N: nat] : groups3539618377306564664at_int(G2,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))),N)))) = groups3539618377306564664at_int(aTP_Lamm_hj(G2),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_6974_sum_Oin__pairs__0,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G2),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))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hk(G2)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_6975_sum_Oin__pairs__0,axiom,
    ! [G2: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(G2),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))),N)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hl(G2)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% sum.in_pairs_0
tff(fact_6976_polyfun__rootbound,axiom,
    ! [C2: fun_nat_complex,K2: nat,N: nat] :
      ( ( aa_nat_complex(C2,K2) != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_qy(C2),N))))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_qy(C2),N)))),N)) ) ) ) ).

% polyfun_rootbound
tff(fact_6977_polyfun__rootbound,axiom,
    ! [C2: fun_nat_real,K2: nat,N: nat] :
      ( ( aa_nat_real(C2,K2) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
       => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_qz(C2),N))))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_real(collect_real(aa_nat_fun_real_bool(aTP_Lamm_qz(C2),N)))),N)) ) ) ) ).

% polyfun_rootbound
tff(fact_6978_polynomial__product,axiom,
    ! [M2: nat,A: fun_nat_complex,N: nat,B: fun_nat_complex,X: complex] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),I3))
         => ( aa_nat_complex(A,I3) = zero_zero_complex ) )
     => ( ! [J3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J3))
           => ( aa_nat_complex(B,J3) = zero_zero_complex ) )
       => ( aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M2))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(B),X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_re(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) ) ) ) ).

% polynomial_product
tff(fact_6979_polynomial__product,axiom,
    ! [M2: nat,A: fun_nat_rat,N: nat,B: fun_nat_rat,X: rat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),I3))
         => ( aa_nat_rat(A,I3) = zero_zero_rat ) )
     => ( ! [J3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J3))
           => ( aa_nat_rat(B,J3) = zero_zero_rat ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_ra(A),X),aa_nat_set_nat(set_ord_atMost_nat,M2))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_ra(B),X),aa_nat_set_nat(set_ord_atMost_nat,N))) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_rg(A),B),X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) ) ) ) ).

% polynomial_product
tff(fact_6980_polynomial__product,axiom,
    ! [M2: nat,A: fun_nat_int,N: nat,B: fun_nat_int,X: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),I3))
         => ( aa_nat_int(A,I3) = zero_zero_int ) )
     => ( ! [J3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J3))
           => ( aa_nat_int(B,J3) = zero_zero_int ) )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rb(A),X),aa_nat_set_nat(set_ord_atMost_nat,M2))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rb(B),X),aa_nat_set_nat(set_ord_atMost_nat,N))) = groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_ri(A),B),X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) ) ) ) ).

% polynomial_product
tff(fact_6981_polynomial__product,axiom,
    ! [M2: nat,A: fun_nat_real,N: nat,B: fun_nat_real,X: real] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),I3))
         => ( aa_nat_real(A,I3) = zero_zero_real ) )
     => ( ! [J3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J3))
           => ( aa_nat_real(B,J3) = 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_qj(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M2))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(B),X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_rk(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) ) ) ) ).

% polynomial_product
tff(fact_6982_prod_Oin__pairs__0,axiom,
    ! [G2: fun_nat_real,N: nat] : aa_set_nat_real(groups129246275422532515t_real(G2),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))),N)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_np(G2)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_6983_prod_Oin__pairs__0,axiom,
    ! [G2: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G2,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))),N)))) = groups73079841787564623at_rat(aTP_Lamm_nq(G2),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_6984_prod_Oin__pairs__0,axiom,
    ! [G2: fun_nat_int,N: nat] : groups705719431365010083at_int(G2,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))),N)))) = groups705719431365010083at_int(aTP_Lamm_nr(G2),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_6985_prod_Oin__pairs__0,axiom,
    ! [G2: fun_nat_nat,N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G2),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))),N)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ns(G2)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% prod.in_pairs_0
tff(fact_6986_polyfun__eq__const,axiom,
    ! [C2: fun_nat_complex,N: nat,K2: complex] :
      ( ! [X4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(C2),X4)),aa_nat_set_nat(set_ord_atMost_nat,N)) = K2
    <=> ( ( aa_nat_complex(C2,zero_zero_nat) = K2 )
        & ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)))
           => ( aa_nat_complex(C2,X4) = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
tff(fact_6987_polyfun__eq__const,axiom,
    ! [C2: fun_nat_real,N: nat,K2: real] :
      ( ! [X4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C2),X4)),aa_nat_set_nat(set_ord_atMost_nat,N)) = K2
    <=> ( ( aa_nat_real(C2,zero_zero_nat) = K2 )
        & ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N)))
           => ( aa_nat_real(C2,X4) = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
tff(fact_6988_binomial__ring,axiom,
    ! [A: complex,B: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(A),B)),N) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_rl(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_6989_binomial__ring,axiom,
    ! [A: rat,B: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),N) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rm(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_6990_binomial__ring,axiom,
    ! [A: int,B: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),N) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rn(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_6991_binomial__ring,axiom,
    ! [A: nat,B: nat,N: 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)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_rc(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_6992_binomial__ring,axiom,
    ! [A: real,B: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(A),B)),N) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ro(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial_ring
tff(fact_6993_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),N) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rp(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_6994_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),N) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rq(A),B),N),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_6995_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(A),B)),N) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_rr(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% pochhammer_binomial_sum
tff(fact_6996_polynomial__product__nat,axiom,
    ! [M2: nat,A: fun_nat_nat,N: nat,B: fun_nat_nat,X: nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),I3))
         => ( aa_nat_nat(A,I3) = zero_zero_nat ) )
     => ( ! [J3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J3))
           => ( aa_nat_nat(B,J3) = 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_rs(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M2))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rs(B),X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ru(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_6997_choose__square__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rv,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = 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))),N)),N) ).

% choose_square_sum
tff(fact_6998_set__update__distinct,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(distinct_VEBT_VEBT,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,N,X)) = insert_VEBT_VEBT(X,minus_5127226145743854075T_VEBT(set_VEBT_VEBT2(Xs),insert_VEBT_VEBT(nth_VEBT_VEBT(Xs,N),bot_bo8194388402131092736T_VEBT))) ) ) ) ).

% set_update_distinct
tff(fact_6999_set__update__distinct,axiom,
    ! [Xs: list_bool,N: nat,X: bool] :
      ( pp(aa_list_bool_bool(distinct_bool,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
       => ( set_bool2(list_update_bool(Xs,N,X)) = insert_bool(X,minus_minus_set_bool(set_bool2(Xs),insert_bool(aa_nat_bool(nth_bool(Xs),N),bot_bot_set_bool))) ) ) ) ).

% set_update_distinct
tff(fact_7000_set__update__distinct,axiom,
    ! [Xs: list_int,N: nat,X: int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
       => ( set_int2(list_update_int(Xs,N,X)) = insert_int(X,minus_minus_set_int(set_int2(Xs),insert_int(aa_nat_int(nth_int(Xs),N),bot_bot_set_int))) ) ) ) ).

% set_update_distinct
tff(fact_7001_set__update__distinct,axiom,
    ! [Xs: list_real,N: nat,X: real] :
      ( distinct_real(Xs)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
       => ( set_real2(list_update_real(Xs,N,X)) = insert_real(X,minus_minus_set_real(set_real2(Xs),insert_real(aa_nat_real(nth_real(Xs),N),bot_bot_set_real))) ) ) ) ).

% set_update_distinct
tff(fact_7002_set__update__distinct,axiom,
    ! [Xs: list_P6011104703257516679at_nat,N: nat,X: product_prod_nat_nat] :
      ( distin6923225563576452346at_nat(Xs)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s5460976970255530739at_nat(Xs)))
       => ( set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,N,X)) = insert8211810215607154385at_nat(X,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,set_Pr5648618587558075414at_nat(Xs)),insert8211810215607154385at_nat(nth_Pr7617993195940197384at_nat(Xs,N),bot_bo2099793752762293965at_nat))) ) ) ) ).

% set_update_distinct
tff(fact_7003_set__update__distinct,axiom,
    ! [Xs: list_nat,N: nat,X: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( set_nat2(list_update_nat(Xs,N,X)) = insert_nat(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,set_nat2(Xs)),insert_nat(aa_nat_nat(nth_nat(Xs),N),bot_bot_set_nat))) ) ) ) ).

% set_update_distinct
tff(fact_7004_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_rw(K2),G2),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_rx(K2),G2),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7005_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ry(K2),G2),H),aa_nat_set_nat(set_ord_atMost_nat,P)) = groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rz(K2),G2),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7006_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: fun_nat_int,H: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_sa(K2),G2),H),aa_nat_set_nat(set_ord_atMost_nat,P)) = groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_sb(K2),G2),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7007_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sc(K2),G2),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sd(K2),G2),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7008_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_se(K2),G2),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_sf(K2),G2),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7009_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: fun_nat_Code_integer,H: fun_nat_Code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( groups3455450783089532116nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_sg(K2),G2),H),aa_nat_set_nat(set_ord_atMost_nat,P)) = groups3455450783089532116nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_sh(K2),G2),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7010_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( aa_set_nat_complex(groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_si(K2),G2),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_complex(groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_rx(K2),G2),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7011_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_sj(K2),G2),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_sf(K2),G2),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7012_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: fun_nat_int,H: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_sk(K2),G2),H),aa_nat_set_nat(set_ord_atMost_nat,P)) = groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_sb(K2),G2),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7013_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),P))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sl(K2),G2),H)),aa_nat_set_nat(set_ord_atMost_nat,P)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sd(K2),G2),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7014_gbinomial__sum__up__index,axiom,
    ! [K2: nat,N: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_sm(K2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_7015_gbinomial__sum__up__index,axiom,
    ! [K2: nat,N: nat] : groups2906978787729119204at_rat(aTP_Lamm_sn(K2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_7016_gbinomial__sum__up__index,axiom,
    ! [K2: nat,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_so,K2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_7017_exp__div__exp__eq,axiom,
    ! [M2: nat,N: 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))),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)),zero_zero_nat))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)))),aa_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,M2),N))) ).

% exp_div_exp_eq
tff(fact_7018_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M2)),zero_zero_int))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)))),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,M2),N))) ).

% exp_div_exp_eq
tff(fact_7019_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] : divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M2)),zero_z3403309356797280102nteger))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ).

% exp_div_exp_eq
tff(fact_7020_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
     => ( aa_nat_complex(gbinomial_complex(A),K2) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7021_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
     => ( aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7022_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(A),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7023_sum__gp0,axiom,
    ! [X: complex,N: nat] :
      ( ( ( X = one_one_complex )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ) )
      & ( ( X != one_one_complex )
       => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(X)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N)))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp0
tff(fact_7024_sum__gp0,axiom,
    ! [X: rat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)) ) ) ) ).

% sum_gp0
tff(fact_7025_sum__gp0,axiom,
    ! [X: real,N: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ) )
      & ( ( X != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N)))),aa_real_real(minus_minus_real(one_one_real),X)) ) ) ) ).

% sum_gp0
tff(fact_7026_polyfun__diff__alt,axiom,
    ! [N: 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),N))
     => ( aa_complex_complex(minus_minus_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(A),X)),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_sq(N),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_7027_polyfun__diff__alt,axiom,
    ! [N: 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),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_ra(A),X),aa_nat_set_nat(set_ord_atMost_nat,N))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_ra(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_ss(N),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_7028_polyfun__diff__alt,axiom,
    ! [N: 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),N))
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rb(A),X),aa_nat_set_nat(set_ord_atMost_nat,N))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rb(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N))) = 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_su(N),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_7029_polyfun__diff__alt,axiom,
    ! [N: 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),N))
     => ( aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(A),X)),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_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_sw(N),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff_alt
tff(fact_7030_binomial__r__part__sum,axiom,
    ! [M2: 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))),M2)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_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))),M2)) ).

% binomial_r_part_sum
tff(fact_7031_choose__linear__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sx,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_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,N),one_one_nat))) ).

% choose_linear_sum
tff(fact_7032_card__lists__length__le,axiom,
    ! [A4: set_list_nat,N: nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( finite7325466520557071688st_nat(collec5989764272469232197st_nat(aa_nat3401988846471794802t_bool(aTP_Lamm_sy(A4),N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_list_nat(A4))),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ) ).

% card_lists_length_le
tff(fact_7033_card__lists__length__le,axiom,
    ! [A4: set_set_nat,N: nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( finite5631907774883551598et_nat(collect_list_set_nat(aa_nat948039967059030092t_bool(aTP_Lamm_sz(A4),N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_set_nat(A4))),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ) ).

% card_lists_length_le
tff(fact_7034_card__lists__length__le,axiom,
    ! [A4: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( finite5120063068150530198omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_be(A4),N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_complex(A4))),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ) ).

% card_lists_length_le
tff(fact_7035_card__lists__length__le,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A4)
     => ( finite5915292604075114978T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_bf(A4),N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,finite7802652506058667612T_VEBT(A4))),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ) ).

% card_lists_length_le
tff(fact_7036_card__lists__length__le,axiom,
    ! [A4: set_bool,N: nat] :
      ( finite_finite_bool(A4)
     => ( finite1699927739564962373t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_bg(A4),N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_bool(A4))),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ) ).

% card_lists_length_le
tff(fact_7037_card__lists__length__le,axiom,
    ! [A4: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( finite_card_list_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_bh(A4),N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_int(A4))),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ) ).

% card_lists_length_le
tff(fact_7038_card__lists__length__le,axiom,
    ! [A4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_bi(A4),N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set_nat_nat(finite_card_nat,A4))),aa_nat_set_nat(set_ord_atMost_nat,N)) ) ) ).

% card_lists_length_le
tff(fact_7039_polyfun__extremal__lemma,axiom,
    ! [E: real,C2: fun_nat_complex,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
     => ? [M8: real] :
        ! [Z4: complex] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M8),aa_complex_real(real_V1022390504157884413omplex,Z4)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(C2),Z4)),aa_nat_set_nat(set_ord_atMost_nat,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z4)),aa_nat_nat(suc,N))))) ) ) ).

% polyfun_extremal_lemma
tff(fact_7040_polyfun__extremal__lemma,axiom,
    ! [E: real,C2: fun_nat_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
     => ? [M8: real] :
        ! [Z4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M8),aa_real_real(real_V7735802525324610683m_real,Z4)))
         => 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_qj(C2),Z4)),aa_nat_set_nat(set_ord_atMost_nat,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Z4)),aa_nat_nat(suc,N))))) ) ) ).

% polyfun_extremal_lemma
tff(fact_7041_polyfun__diff,axiom,
    ! [N: 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),N))
     => ( aa_complex_complex(minus_minus_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(A),X)),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_tb(N),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_7042_polyfun__diff,axiom,
    ! [N: 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),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_ra(A),X),aa_nat_set_nat(set_ord_atMost_nat,N))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_ra(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_td(N),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_7043_polyfun__diff,axiom,
    ! [N: 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),N))
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rb(A),X),aa_nat_set_nat(set_ord_atMost_nat,N))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rb(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N))) = 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_tf(N),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_7044_polyfun__diff,axiom,
    ! [N: 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),N))
     => ( aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(A),X)),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_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_th(N),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ) ) ).

% polyfun_diff
tff(fact_7045_card__lists__distinct__length__eq_H,axiom,
    ! [K2: nat,A4: set_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),finite_card_complex(A4)))
     => ( finite5120063068150530198omplex(collect_list_complex(aa_set1925655978115286672x_bool(aTP_Lamm_ti(K2),A4))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A4)),K2)),one_one_nat)),finite_card_complex(A4))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_7046_card__lists__distinct__length__eq_H,axiom,
    ! [K2: nat,A4: set_list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),finite_card_list_nat(A4)))
     => ( finite7325466520557071688st_nat(collec5989764272469232197st_nat(aa_set3810083887808699692t_bool(aTP_Lamm_tj(K2),A4))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A4)),K2)),one_one_nat)),finite_card_list_nat(A4))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_7047_card__lists__distinct__length__eq_H,axiom,
    ! [K2: nat,A4: set_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),finite_card_set_nat(A4)))
     => ( finite5631907774883551598et_nat(collect_list_set_nat(aa_set736033600575934688t_bool(aTP_Lamm_tk(K2),A4))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A4)),K2)),one_one_nat)),finite_card_set_nat(A4))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_7048_card__lists__distinct__length__eq_H,axiom,
    ! [K2: nat,A4: set_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),finite7802652506058667612T_VEBT(A4)))
     => ( finite5915292604075114978T_VEBT(collec5608196760682091941T_VEBT(aa_set3008069279726893048T_bool(aTP_Lamm_tl(K2),A4))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite7802652506058667612T_VEBT(A4)),K2)),one_one_nat)),finite7802652506058667612T_VEBT(A4))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_7049_card__lists__distinct__length__eq_H,axiom,
    ! [K2: nat,A4: set_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),finite_card_bool(A4)))
     => ( finite1699927739564962373t_bool(collect_list_bool(aa_set1157894234898225080l_bool(aTP_Lamm_tm(K2),A4))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_bool(A4)),K2)),one_one_nat)),finite_card_bool(A4))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_7050_card__lists__distinct__length__eq_H,axiom,
    ! [K2: nat,A4: set_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),finite_card_int(A4)))
     => ( finite_card_list_int(collect_list_int(aa_set9099777061830656916t_bool(aTP_Lamm_tn(K2),A4))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A4)),K2)),one_one_nat)),finite_card_int(A4))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_7051_card__lists__distinct__length__eq_H,axiom,
    ! [K2: nat,A4: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),aa_set_nat_nat(finite_card_nat,A4)))
     => ( finite_card_list_nat(collect_list_nat(aa_set1433770957266526284t_bool(aTP_Lamm_to(K2),A4))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,A4)),K2)),one_one_nat)),aa_set_nat_nat(finite_card_nat,A4))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_7052_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R: int] : adjust_div(product_Pair_int_int(Q2,R)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q2),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,R),zero_zero_int)))) ).

% Divides.adjust_div_eq
tff(fact_7053_list__decode_Opinduct,axiom,
    ! [A0: nat,P2: 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(P2,zero_zero_nat)) )
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N3)))
             => ( ! [X5: nat,Y4: nat] :
                    ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y4) = aa_nat7603221163087113075at_nat(nat_prod_decode,N3) )
                   => pp(aa_nat_bool(P2,Y4)) )
               => pp(aa_nat_bool(P2,aa_nat_nat(suc,N3))) ) )
         => pp(aa_nat_bool(P2,A0)) ) ) ) ).

% list_decode.pinduct
tff(fact_7054_gbinomial__code,axiom,
    ! [K2: nat,A: complex] :
      ( ( ( K2 = zero_zero_nat )
       => ( aa_nat_complex(gbinomial_complex(A),K2) = one_one_complex ) )
      & ( ( K2 != zero_zero_nat )
       => ( aa_nat_complex(gbinomial_complex(A),K2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_tp(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat)),one_one_complex)),aa_nat_complex(semiri5044797733671781792omplex,K2)) ) ) ) ).

% gbinomial_code
tff(fact_7055_gbinomial__code,axiom,
    ! [K2: nat,A: rat] :
      ( ( ( K2 = zero_zero_nat )
       => ( aa_nat_rat(gbinomial_rat(A),K2) = one_one_rat ) )
      & ( ( K2 != zero_zero_nat )
       => ( aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_tq(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat)),one_one_rat)),aa_nat_rat(semiri773545260158071498ct_rat,K2)) ) ) ) ).

% gbinomial_code
tff(fact_7056_gbinomial__code,axiom,
    ! [K2: nat,A: real] :
      ( ( ( K2 = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = one_one_real ) )
      & ( ( K2 != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_tr(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat)),one_one_real)),aa_nat_real(semiri2265585572941072030t_real,K2)) ) ) ) ).

% gbinomial_code
tff(fact_7057_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_ts(N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex ) ) ).

% choose_alternating_sum
tff(fact_7058_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( groups7501900531339628137nteger(aTP_Lamm_tt(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
tff(fact_7059_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( groups2906978787729119204at_rat(aTP_Lamm_tu(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat ) ) ).

% choose_alternating_sum
tff(fact_7060_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_tv,N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_7061_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_tw,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real ) ) ).

% choose_alternating_sum
tff(fact_7062_signed__take__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_7063_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_7064_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_7065_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_7066_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_7067_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : uminus1351360451143612070nteger(uminus1351360451143612070nteger(A)) = A ).

% add.inverse_inverse
tff(fact_7068_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_7069_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_7070_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_7071_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_7072_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = uminus1351360451143612070nteger(B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_7073_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_7074_of__nat__id,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = N ).

% of_nat_id
tff(fact_7075_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_7076_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_7077_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(B)),uminus1351360451143612070nteger(A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).

% neg_le_iff_le
tff(fact_7078_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_7079_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_7080_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_7081_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_7082_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_7083_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_7084_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_7085_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_7086_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = uminus1351360451143612070nteger(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_7087_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_7088_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_7089_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_7090_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_7091_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_7092_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_7093_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_7094_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_7095_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_7096_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = uminus1351360451143612070nteger(A) )
    <=> ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_7097_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_7098_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_7099_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_7100_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_7101_add_Oinverse__neutral,axiom,
    uminus1351360451143612070nteger(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_7102_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_7103_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_7104_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_7105_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(B)),uminus1351360451143612070nteger(A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).

% neg_less_iff_less
tff(fact_7106_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_7107_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
    <=> ( M2 = N ) ) ).

% neg_numeral_eq_iff
tff(fact_7108_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) )
    <=> ( M2 = N ) ) ).

% neg_numeral_eq_iff
tff(fact_7109_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) )
    <=> ( M2 = N ) ) ).

% neg_numeral_eq_iff
tff(fact_7110_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( uminus1351360451143612070nteger(numera6620942414471956472nteger(M2)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(N)) )
    <=> ( M2 = N ) ) ).

% neg_numeral_eq_iff
tff(fact_7111_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) )
    <=> ( M2 = N ) ) ).

% neg_numeral_eq_iff
tff(fact_7112_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_7113_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_7114_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_7115_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_right
tff(fact_7116_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).

% mult_minus_right
tff(fact_7117_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_7118_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_7119_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_7120_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,uminus1351360451143612070nteger(A)),uminus1351360451143612070nteger(B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) ).

% minus_mult_minus
tff(fact_7121_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) ).

% minus_mult_minus
tff(fact_7122_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_7123_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_7124_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_7125_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,uminus1351360451143612070nteger(A)),B) = uminus1351360451143612070nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_left
tff(fact_7126_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).

% mult_minus_left
tff(fact_7127_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_7128_add__minus__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),B)) = B ).

% add_minus_cancel
tff(fact_7129_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_7130_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),B)) = B ).

% add_minus_cancel
tff(fact_7131_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).

% add_minus_cancel
tff(fact_7132_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_7133_minus__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(plus_plus_real(A),B)) = B ).

% minus_add_cancel
tff(fact_7134_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_7135_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),plus_p5714425477246183910nteger(A,B)) = B ).

% minus_add_cancel
tff(fact_7136_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = B ).

% minus_add_cancel
tff(fact_7137_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_7138_minus__add__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) ).

% minus_add_distrib
tff(fact_7139_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_7140_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] : uminus1351360451143612070nteger(plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),uminus1351360451143612070nteger(B)) ).

% minus_add_distrib
tff(fact_7141_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_add_distrib
tff(fact_7142_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_7143_minus__diff__eq,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(minus_minus_real(A),B)) = aa_real_real(minus_minus_real(B),A) ).

% minus_diff_eq
tff(fact_7144_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_7145_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] : uminus1351360451143612070nteger(minus_8373710615458151222nteger(A,B)) = minus_8373710615458151222nteger(B,A) ).

% minus_diff_eq
tff(fact_7146_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A) ).

% minus_diff_eq
tff(fact_7147_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_7148_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_7149_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_7150_dvd__minus__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),uminus1351360451143612070nteger(Y)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y)) ) ).

% dvd_minus_iff
tff(fact_7151_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_7152_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_7153_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_7154_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_7155_minus__dvd__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,uminus1351360451143612070nteger(X)),Y))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y)) ) ).

% minus_dvd_iff
tff(fact_7156_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_7157_negative__eq__positive,axiom,
    ! [N: nat,M2: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,M2) )
    <=> ( ( N = zero_zero_nat )
        & ( M2 = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_7158_signed__take__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(N),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_7159_set__decode__inverse,axiom,
    ! [N: nat] : aa_set_nat_nat(nat_set_encode,aa_nat_set_nat(nat_set_decode,N)) = N ).

% set_decode_inverse
tff(fact_7160_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_7161_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_eq_nonneg
tff(fact_7162_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_7163_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_7164_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_7165_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),uminus1351360451143612070nteger(A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% less_eq_neg_nonpos
tff(fact_7166_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_7167_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_7168_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_7169_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_le_0_iff_le
tff(fact_7170_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_7171_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_7172_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_7173_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% neg_0_le_iff_le
tff(fact_7174_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_7175_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_7176_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_7177_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_7178_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),uminus1351360451143612070nteger(A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% less_neg_neg
tff(fact_7179_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_7180_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_7181_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_7182_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_pos
tff(fact_7183_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_7184_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_7185_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_7186_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% neg_0_less_iff_less
tff(fact_7187_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_7188_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_7189_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_7190_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_0_iff_less
tff(fact_7191_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_7192_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_7193_add_Oright__inverse,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,A)) = zero_zero_real ).

% add.right_inverse
tff(fact_7194_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_7195_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,uminus1351360451143612070nteger(A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_7196_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_7197_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_7198_ab__left__minus,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_left_minus
tff(fact_7199_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_7200_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_7201_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_7202_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_7203_verit__minus__simplify_I3_J,axiom,
    ! [B: real] : aa_real_real(minus_minus_real(zero_zero_real),B) = aa_real_real(uminus_uminus_real,B) ).

% verit_minus_simplify(3)
tff(fact_7204_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_7205_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = uminus1351360451143612070nteger(B) ).

% verit_minus_simplify(3)
tff(fact_7206_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% verit_minus_simplify(3)
tff(fact_7207_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_7208_diff__0,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ).

% diff_0
tff(fact_7209_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_7210_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = uminus1351360451143612070nteger(A) ).

% diff_0
tff(fact_7211_diff__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ).

% diff_0
tff(fact_7212_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N))) ).

% add_neg_numeral_simps(3)
tff(fact_7213_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,M2)),aa_num_real(numeral_numeral_real,N))) ).

% add_neg_numeral_simps(3)
tff(fact_7214_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M2)),numera6690914467698888265omplex(N))) ).

% add_neg_numeral_simps(3)
tff(fact_7215_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M2)),uminus1351360451143612070nteger(numera6620942414471956472nteger(N))) = uminus1351360451143612070nteger(plus_p5714425477246183910nteger(numera6620942414471956472nteger(M2),numera6620942414471956472nteger(N))) ).

% add_neg_numeral_simps(3)
tff(fact_7216_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_num_rat(numeral_numeral_rat,N))) ).

% add_neg_numeral_simps(3)
tff(fact_7217_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_7218_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_7219_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_7220_mult__minus1__right,axiom,
    ! [Z2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z2),uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(Z2) ).

% mult_minus1_right
tff(fact_7221_mult__minus1__right,axiom,
    ! [Z2: rat] : aa_rat_rat(aa_rat_fun_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_7222_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_7223_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_7224_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_7225_mult__minus1,axiom,
    ! [Z2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,uminus1351360451143612070nteger(one_one_Code_integer)),Z2) = uminus1351360451143612070nteger(Z2) ).

% mult_minus1
tff(fact_7226_mult__minus1,axiom,
    ! [Z2: rat] : aa_rat_rat(aa_rat_fun_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_7227_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_7228_diff__minus__eq__add,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(plus_plus_real(A),B) ).

% diff_minus_eq_add
tff(fact_7229_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_7230_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,uminus1351360451143612070nteger(B)) = plus_p5714425477246183910nteger(A,B) ).

% diff_minus_eq_add
tff(fact_7231_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) ).

% diff_minus_eq_add
tff(fact_7232_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_7233_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(minus_minus_real(B),A) ).

% uminus_add_conv_diff
tff(fact_7234_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_7235_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),B) = minus_8373710615458151222nteger(B,A) ).

% uminus_add_conv_diff
tff(fact_7236_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A) ).

% uminus_add_conv_diff
tff(fact_7237_fact__0,axiom,
    aa_nat_Code_integer(semiri3624122377584611663nteger,zero_zero_nat) = one_one_Code_integer ).

% fact_0
tff(fact_7238_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_7239_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_7240_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_7241_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_7242_negative__zless,axiom,
    ! [N: nat,M2: 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,N)))),aa_nat_int(semiri1314217659103216013at_int,M2))) ).

% negative_zless
tff(fact_7243_signed__take__bit__Suc__1,axiom,
    ! [N: nat] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,N),one_one_Code_integer) = one_one_Code_integer ).

% signed_take_bit_Suc_1
tff(fact_7244_signed__take__bit__Suc__1,axiom,
    ! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_7245_dbl__simps_I1_J,axiom,
    ! [K2: num] : aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K2))) ).

% dbl_simps(1)
tff(fact_7246_dbl__simps_I1_J,axiom,
    ! [K2: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K2))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K2))) ).

% dbl_simps(1)
tff(fact_7247_dbl__simps_I1_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K2))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K2))) ).

% dbl_simps(1)
tff(fact_7248_dbl__simps_I1_J,axiom,
    ! [K2: num] : neg_nu8804712462038260780nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(K2))) = uminus1351360451143612070nteger(neg_nu8804712462038260780nteger(numera6620942414471956472nteger(K2))) ).

% dbl_simps(1)
tff(fact_7249_dbl__simps_I1_J,axiom,
    ! [K2: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K2))) ).

% dbl_simps(1)
tff(fact_7250_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_7251_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_7252_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_7253_dbl__inc__simps_I4_J,axiom,
    neg_nu5831290666863070958nteger(uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% dbl_inc_simps(4)
tff(fact_7254_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_7255_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_7256_set__encode__inverse,axiom,
    ! [A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( aa_nat_set_nat(nat_set_decode,aa_set_nat_nat(nat_set_encode,A4)) = A4 ) ) ).

% set_encode_inverse
tff(fact_7257_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_7258_add__neg__numeral__special_I7_J,axiom,
    aa_real_real(plus_plus_real(one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% add_neg_numeral_special(7)
tff(fact_7259_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_7260_add__neg__numeral__special_I7_J,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,uminus1351360451143612070nteger(one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_7261_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(aa_rat_fun_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_7262_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_7263_add__neg__numeral__special_I8_J,axiom,
    aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = zero_zero_real ).

% add_neg_numeral_special(8)
tff(fact_7264_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_7265_add__neg__numeral__special_I8_J,axiom,
    plus_p5714425477246183910nteger(uminus1351360451143612070nteger(one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_7266_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(aa_rat_fun_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_7267_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_7268_diff__numeral__special_I12_J,axiom,
    aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% diff_numeral_special(12)
tff(fact_7269_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_7270_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(uminus1351360451143612070nteger(one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_7271_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(aa_rat_fun_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_7272_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_7273_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_7274_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_7275_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( uminus1351360451143612070nteger(one_one_Code_integer) = uminus1351360451143612070nteger(numera6620942414471956472nteger(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_7276_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_7277_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_7278_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_7279_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_7280_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( uminus1351360451143612070nteger(numera6620942414471956472nteger(N)) = uminus1351360451143612070nteger(one_one_Code_integer) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_7281_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_7282_left__minus__one__mult__self,axiom,
    ! [N: 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)),N)),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)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_7283_left__minus__one__mult__self,axiom,
    ! [N: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_7284_left__minus__one__mult__self,axiom,
    ! [N: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_7285_left__minus__one__mult__self,axiom,
    ! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_7286_left__minus__one__mult__self,axiom,
    ! [N: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_7287_minus__one__mult__self,axiom,
    ! [N: 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)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)) = one_one_int ).

% minus_one_mult_self
tff(fact_7288_minus__one__mult__self,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)) = one_one_real ).

% minus_one_mult_self
tff(fact_7289_minus__one__mult__self,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)) = one_one_complex ).

% minus_one_mult_self
tff(fact_7290_minus__one__mult__self,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),N)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_7291_minus__one__mult__self,axiom,
    ! [N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)) = one_one_rat ).

% minus_one_mult_self
tff(fact_7292_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_7293_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,uminus1351360451143612070nteger(one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_7294_max__number__of_I2_J,axiom,
    ! [U: num,V2: 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,V2))))
       => ( aa_real_real(ord_max_real(aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2)) ) )
      & ( ~ 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,V2))))
       => ( aa_real_real(ord_max_real(aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))) = aa_num_real(numeral_numeral_real,U) ) ) ) ).

% max_number_of(2)
tff(fact_7295_max__number__of_I2_J,axiom,
    ! [U: num,V2: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),uminus1351360451143612070nteger(numera6620942414471956472nteger(V2))))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),uminus1351360451143612070nteger(numera6620942414471956472nteger(V2))) = uminus1351360451143612070nteger(numera6620942414471956472nteger(V2)) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),uminus1351360451143612070nteger(numera6620942414471956472nteger(V2))))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),uminus1351360451143612070nteger(numera6620942414471956472nteger(V2))) = numera6620942414471956472nteger(U) ) ) ) ).

% max_number_of(2)
tff(fact_7296_max__number__of_I2_J,axiom,
    ! [U: num,V2: 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,V2))))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2)) ) )
      & ( ~ 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,V2))))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2))) = aa_num_rat(numeral_numeral_rat,U) ) ) ) ).

% max_number_of(2)
tff(fact_7297_max__number__of_I2_J,axiom,
    ! [U: num,V2: 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,V2))))
       => ( 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,V2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2)) ) )
      & ( ~ 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,V2))))
       => ( 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,V2))) = aa_num_int(numeral_numeral_int,U) ) ) ) ).

% max_number_of(2)
tff(fact_7298_max__number__of_I3_J,axiom,
    ! [U: num,V2: 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,V2)))
       => ( aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V2)) = aa_num_real(numeral_numeral_real,V2) ) )
      & ( ~ 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,V2)))
       => ( aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V2)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U)) ) ) ) ).

% max_number_of(3)
tff(fact_7299_max__number__of_I3_J,axiom,
    ! [U: num,V2: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V2)))
       => ( ord_max_Code_integer(uminus1351360451143612070nteger(numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V2)) = numera6620942414471956472nteger(V2) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V2)))
       => ( ord_max_Code_integer(uminus1351360451143612070nteger(numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V2)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(U)) ) ) ) ).

% max_number_of(3)
tff(fact_7300_max__number__of_I3_J,axiom,
    ! [U: num,V2: 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,V2)))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V2)) = aa_num_rat(numeral_numeral_rat,V2) ) )
      & ( ~ 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,V2)))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V2)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U)) ) ) ) ).

% max_number_of(3)
tff(fact_7301_max__number__of_I3_J,axiom,
    ! [U: num,V2: 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,V2)))
       => ( 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,V2)) = aa_num_int(numeral_numeral_int,V2) ) )
      & ( ~ 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,V2)))
       => ( 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,V2)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U)) ) ) ) ).

% max_number_of(3)
tff(fact_7302_max__number__of_I4_J,axiom,
    ! [U: num,V2: 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,V2))))
       => ( aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2)) ) )
      & ( ~ 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,V2))))
       => ( aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U)) ) ) ) ).

% max_number_of(4)
tff(fact_7303_max__number__of_I4_J,axiom,
    ! [U: num,V2: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(U))),uminus1351360451143612070nteger(numera6620942414471956472nteger(V2))))
       => ( ord_max_Code_integer(uminus1351360451143612070nteger(numera6620942414471956472nteger(U)),uminus1351360451143612070nteger(numera6620942414471956472nteger(V2))) = uminus1351360451143612070nteger(numera6620942414471956472nteger(V2)) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(U))),uminus1351360451143612070nteger(numera6620942414471956472nteger(V2))))
       => ( ord_max_Code_integer(uminus1351360451143612070nteger(numera6620942414471956472nteger(U)),uminus1351360451143612070nteger(numera6620942414471956472nteger(V2))) = uminus1351360451143612070nteger(numera6620942414471956472nteger(U)) ) ) ) ).

% max_number_of(4)
tff(fact_7304_max__number__of_I4_J,axiom,
    ! [U: num,V2: 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,V2))))
       => ( aa_rat_rat(aa_rat_fun_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,V2))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2)) ) )
      & ( ~ 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,V2))))
       => ( aa_rat_rat(aa_rat_fun_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,V2))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U)) ) ) ) ).

% max_number_of(4)
tff(fact_7305_max__number__of_I4_J,axiom,
    ! [U: num,V2: 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,V2))))
       => ( 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,V2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2)) ) )
      & ( ~ 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,V2))))
       => ( 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,V2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U)) ) ) ) ).

% max_number_of(4)
tff(fact_7306_fact__Suc__0,axiom,
    aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,zero_zero_nat)) = one_one_Code_integer ).

% fact_Suc_0
tff(fact_7307_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_7308_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_7309_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_7310_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_7311_fact__Suc,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).

% fact_Suc
tff(fact_7312_fact__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))),aa_nat_int(semiri1406184849735516958ct_int,N)) ).

% fact_Suc
tff(fact_7313_fact__Suc,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N))),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ).

% fact_Suc
tff(fact_7314_fact__Suc,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))),aa_nat_real(semiri2265585572941072030t_real,N)) ).

% fact_Suc
tff(fact_7315_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N))) ).

% diff_numeral_simps(3)
tff(fact_7316_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_num_real(numeral_numeral_real,N)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N))) ).

% diff_numeral_simps(3)
tff(fact_7317_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2))),numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N))) ).

% diff_numeral_simps(3)
tff(fact_7318_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M2)),numera6620942414471956472nteger(N)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N))) ).

% diff_numeral_simps(3)
tff(fact_7319_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N))) ).

% diff_numeral_simps(3)
tff(fact_7320_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)) ).

% diff_numeral_simps(2)
tff(fact_7321_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] : aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)) ).

% diff_numeral_simps(2)
tff(fact_7322_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M2)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)) ).

% diff_numeral_simps(2)
tff(fact_7323_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M2),uminus1351360451143612070nteger(numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)) ).

% diff_numeral_simps(2)
tff(fact_7324_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)) ).

% diff_numeral_simps(2)
tff(fact_7325_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_7326_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_7327_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M2)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M2),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_7328_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(M2)),uminus1351360451143612070nteger(numera6620942414471956472nteger(N))) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,M2),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_7329_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_7330_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_7331_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_num_real(numeral_numeral_real,N)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_7332_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2))),numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M2),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_7333_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M2))),numera6620942414471956472nteger(N)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,M2),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_7334_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_num_rat(numeral_numeral_rat,N)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_7335_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_7336_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_7337_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M2),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_7338_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M2))),uminus1351360451143612070nteger(numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,M2),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_7339_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_7340_semiring__norm_I172_J,axiom,
    ! [V2: num,W2: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W2))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2))),Y) ).

% semiring_norm(172)
tff(fact_7341_semiring__norm_I172_J,axiom,
    ! [V2: num,W2: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2))),Y) ).

% semiring_norm(172)
tff(fact_7342_semiring__norm_I172_J,axiom,
    ! [V2: num,W2: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V2))),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),Y)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2))),Y) ).

% semiring_norm(172)
tff(fact_7343_semiring__norm_I172_J,axiom,
    ! [V2: num,W2: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(V2))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(W2))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2))),Y) ).

% semiring_norm(172)
tff(fact_7344_semiring__norm_I172_J,axiom,
    ! [V2: num,W2: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2))),Y) ).

% semiring_norm(172)
tff(fact_7345_semiring__norm_I171_J,axiom,
    ! [V2: num,W2: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W2))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2)))),Y) ).

% semiring_norm(171)
tff(fact_7346_semiring__norm_I171_J,axiom,
    ! [V2: num,W2: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2)))),Y) ).

% semiring_norm(171)
tff(fact_7347_semiring__norm_I171_J,axiom,
    ! [V2: num,W2: num,Y: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V2)),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2)))),Y) ).

% semiring_norm(171)
tff(fact_7348_semiring__norm_I171_J,axiom,
    ! [V2: num,W2: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(W2))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2)))),Y) ).

% semiring_norm(171)
tff(fact_7349_semiring__norm_I171_J,axiom,
    ! [V2: num,W2: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2)))),Y) ).

% semiring_norm(171)
tff(fact_7350_semiring__norm_I170_J,axiom,
    ! [V2: num,W2: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W2)),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2)))),Y) ).

% semiring_norm(170)
tff(fact_7351_semiring__norm_I170_J,axiom,
    ! [V2: num,W2: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2)))),Y) ).

% semiring_norm(170)
tff(fact_7352_semiring__norm_I170_J,axiom,
    ! [V2: num,W2: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V2))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2)))),Y) ).

% semiring_norm(170)
tff(fact_7353_semiring__norm_I170_J,axiom,
    ! [V2: num,W2: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(V2))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W2)),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2)))),Y) ).

% semiring_norm(170)
tff(fact_7354_semiring__norm_I170_J,axiom,
    ! [V2: num,W2: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V2),W2)))),Y) ).

% semiring_norm(170)
tff(fact_7355_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M2)) ) ).

% neg_numeral_le_iff
tff(fact_7356_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M2))),uminus1351360451143612070nteger(numera6620942414471956472nteger(N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M2)) ) ).

% neg_numeral_le_iff
tff(fact_7357_neg__numeral__le__iff,axiom,
    ! [M2: num,N: 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,M2))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M2)) ) ).

% neg_numeral_le_iff
tff(fact_7358_neg__numeral__le__iff,axiom,
    ! [M2: num,N: 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,M2))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M2)) ) ).

% neg_numeral_le_iff
tff(fact_7359_neg__numeral__less__iff,axiom,
    ! [M2: num,N: 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,M2))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M2)) ) ).

% neg_numeral_less_iff
tff(fact_7360_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M2)) ) ).

% neg_numeral_less_iff
tff(fact_7361_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M2))),uminus1351360451143612070nteger(numera6620942414471956472nteger(N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M2)) ) ).

% neg_numeral_less_iff
tff(fact_7362_neg__numeral__less__iff,axiom,
    ! [M2: num,N: 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,M2))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M2)) ) ).

% neg_numeral_less_iff
tff(fact_7363_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_7364_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_7365_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_7366_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_7367_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_7368_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: 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,M2))))
    <=> ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_7369_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),uminus1351360451143612070nteger(numera6620942414471956472nteger(M2))))
    <=> ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_7370_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: 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,M2))))
    <=> ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_7371_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: 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,M2))))
    <=> ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_7372_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(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_7373_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(aa_rat_fun_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(aa_rat_fun_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_7374_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(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_7375_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(aa_rat_fun_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(aa_rat_fun_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_7376_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( aa_real_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_7377_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_7378_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_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(aa_rat_fun_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_7379_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A = aa_real_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_7380_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_7381_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A = aa_rat_rat(aa_rat_fun_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(aa_rat_fun_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_7382_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: 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,M2))),aa_int_int(uminus_uminus_int,one_one_int)))
    <=> ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_7383_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,one_one_real)))
    <=> ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_7384_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M2))),uminus1351360451143612070nteger(one_one_Code_integer)))
    <=> ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_7385_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: 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,M2))),aa_rat_rat(uminus_uminus_rat,one_one_rat)))
    <=> ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_7386_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(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_7387_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(aa_rat_fun_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(aa_rat_fun_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_7388_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(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_7389_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(aa_rat_fun_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(aa_rat_fun_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_7390_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K2))) ).

% dbl_dec_simps(1)
tff(fact_7391_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K2))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K2))) ).

% dbl_dec_simps(1)
tff(fact_7392_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K2))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K2))) ).

% dbl_dec_simps(1)
tff(fact_7393_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : neg_nu7757733837767384882nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(K2))) = uminus1351360451143612070nteger(neg_nu5831290666863070958nteger(numera6620942414471956472nteger(K2))) ).

% dbl_dec_simps(1)
tff(fact_7394_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K2))) ).

% dbl_dec_simps(1)
tff(fact_7395_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K2))) ).

% dbl_inc_simps(1)
tff(fact_7396_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K2))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K2))) ).

% dbl_inc_simps(1)
tff(fact_7397_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K2))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K2))) ).

% dbl_inc_simps(1)
tff(fact_7398_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : neg_nu5831290666863070958nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(K2))) = uminus1351360451143612070nteger(neg_nu7757733837767384882nteger(numera6620942414471956472nteger(K2))) ).

% dbl_inc_simps(1)
tff(fact_7399_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K2))) ).

% dbl_inc_simps(1)
tff(fact_7400_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_7401_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_bit0
tff(fact_7402_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_7403_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_7404_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_7405_add__neg__numeral__special_I9_J,axiom,
    plus_p5714425477246183910nteger(uminus1351360451143612070nteger(one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_7406_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(aa_rat_fun_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_7407_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_7408_diff__numeral__special_I11_J,axiom,
    aa_real_real(minus_minus_real(one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_7409_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_7410_diff__numeral__special_I11_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,uminus1351360451143612070nteger(one_one_Code_integer)) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_7411_diff__numeral__special_I11_J,axiom,
    aa_rat_rat(aa_rat_fun_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_7412_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_7413_diff__numeral__special_I10_J,axiom,
    aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_7414_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_7415_diff__numeral__special_I10_J,axiom,
    minus_8373710615458151222nteger(uminus1351360451143612070nteger(one_one_Code_integer),one_one_Code_integer) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_7416_diff__numeral__special_I10_J,axiom,
    aa_rat_rat(aa_rat_fun_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_7417_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = 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))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_7418_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_7419_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_7420_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_7421_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = 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))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_7422_diff__numeral__special_I4_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),one))) ).

% diff_numeral_special(4)
tff(fact_7423_diff__numeral__special_I4_J,axiom,
    ! [M2: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),one))) ).

% diff_numeral_special(4)
tff(fact_7424_diff__numeral__special_I4_J,axiom,
    ! [M2: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),one))) ).

% diff_numeral_special(4)
tff(fact_7425_diff__numeral__special_I4_J,axiom,
    ! [M2: num] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M2)),one_one_Code_integer) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),one))) ).

% diff_numeral_special(4)
tff(fact_7426_diff__numeral__special_I4_J,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),one))) ).

% diff_numeral_special(4)
tff(fact_7427_diff__numeral__special_I3_J,axiom,
    ! [N: 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,N))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_7428_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_real_real(minus_minus_real(one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_7429_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_7430_diff__numeral__special_I3_J,axiom,
    ! [N: num] : minus_8373710615458151222nteger(one_one_Code_integer,uminus1351360451143612070nteger(numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_7431_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_7432_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,N)),aa_nat_set_nat(nat_set_decode,X)))
    <=> pp(aa_set_nat_bool(member_nat(N),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_7433_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_7434_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_7435_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_7436_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_7437_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_7438_power__minus1__even,axiom,
    ! [N: 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))),N)) = one_one_int ).

% power_minus1_even
tff(fact_7439_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_real ).

% power_minus1_even
tff(fact_7440_power__minus1__even,axiom,
    ! [N: 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))),N)) = one_one_complex ).

% power_minus1_even
tff(fact_7441_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(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))),N)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_7442_power__minus1__even,axiom,
    ! [N: 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))),N)) = one_one_rat ).

% power_minus1_even
tff(fact_7443_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = uminus1351360451143612070nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_7444_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_ri631733984087533419it_int(zero_zero_nat),A) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_7445_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_7446_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_7447_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_7448_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = uminus1351360451143612070nteger(B) )
    <=> ( B = uminus1351360451143612070nteger(A) ) ) ).

% equation_minus_iff
tff(fact_7449_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_7450_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_7451_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_7452_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_7453_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = B )
    <=> ( uminus1351360451143612070nteger(B) = A ) ) ).

% minus_equation_iff
tff(fact_7454_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_7455_fact__mono__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).

% fact_mono_nat
tff(fact_7456_fact__ge__self,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_self
tff(fact_7457_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_complex(semiri5044797733671781792omplex,N) != zero_zero_complex ).

% fact_nonzero
tff(fact_7458_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N) != zero_zero_rat ).

% fact_nonzero
tff(fact_7459_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) != zero_zero_int ).

% fact_nonzero
tff(fact_7460_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) != zero_zero_nat ).

% fact_nonzero
tff(fact_7461_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,N) != zero_zero_real ).

% fact_nonzero
tff(fact_7462_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_7463_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_7464_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_7465_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_7466_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,uminus1351360451143612070nteger(B)),uminus1351360451143612070nteger(A))) ) ).

% le_imp_neg_le
tff(fact_7467_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_7468_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_7469_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_7470_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(B)),A)) ) ).

% minus_le_iff
tff(fact_7471_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_7472_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_7473_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_7474_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),uminus1351360451143612070nteger(B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),uminus1351360451143612070nteger(A))) ) ).

% le_minus_iff
tff(fact_7475_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_7476_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_7477_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_7478_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_7479_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),uminus1351360451143612070nteger(B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),uminus1351360451143612070nteger(A))) ) ).

% less_minus_iff
tff(fact_7480_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_7481_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_7482_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_7483_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(B)),A)) ) ).

% minus_less_iff
tff(fact_7484_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_7485_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2)) != aa_num_int(numeral_numeral_int,N) ).

% neg_numeral_neq_numeral
tff(fact_7486_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2)) != aa_num_real(numeral_numeral_real,N) ).

% neg_numeral_neq_numeral
tff(fact_7487_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M2)) != numera6690914467698888265omplex(N) ).

% neg_numeral_neq_numeral
tff(fact_7488_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] : uminus1351360451143612070nteger(numera6620942414471956472nteger(M2)) != numera6620942414471956472nteger(N) ).

% neg_numeral_neq_numeral
tff(fact_7489_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M2)) != aa_num_rat(numeral_numeral_rat,N) ).

% neg_numeral_neq_numeral
tff(fact_7490_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] : aa_num_int(numeral_numeral_int,M2) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% numeral_neq_neg_numeral
tff(fact_7491_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] : aa_num_real(numeral_numeral_real,M2) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) ).

% numeral_neq_neg_numeral
tff(fact_7492_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] : numera6690914467698888265omplex(M2) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% numeral_neq_neg_numeral
tff(fact_7493_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] : numera6620942414471956472nteger(M2) != uminus1351360451143612070nteger(numera6620942414471956472nteger(N)) ).

% numeral_neq_neg_numeral
tff(fact_7494_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] : aa_num_rat(numeral_numeral_rat,M2) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).

% numeral_neq_neg_numeral
tff(fact_7495_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_7496_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_7497_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_7498_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,uminus1351360451143612070nteger(A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),uminus1351360451143612070nteger(B)) ).

% minus_mult_commute
tff(fact_7499_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_mult_commute
tff(fact_7500_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_7501_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_7502_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_7503_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 = uminus1351360451143612070nteger(B) ) ) ) ).

% square_eq_iff
tff(fact_7504_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),B) )
    <=> ( ( A = B )
        | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% square_eq_iff
tff(fact_7505_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_7506_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% is_num_normalize(8)
tff(fact_7507_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_7508_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] : uminus1351360451143612070nteger(plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(B),uminus1351360451143612070nteger(A)) ).

% is_num_normalize(8)
tff(fact_7509_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% is_num_normalize(8)
tff(fact_7510_group__cancel_Oneg1,axiom,
    ! [A4: int,K2: int,A: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(uminus_uminus_int,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K2)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_7511_group__cancel_Oneg1,axiom,
    ! [A4: real,K2: real,A: real] :
      ( ( A4 = aa_real_real(plus_plus_real(K2),A) )
     => ( aa_real_real(uminus_uminus_real,A4) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,K2)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_7512_group__cancel_Oneg1,axiom,
    ! [A4: complex,K2: complex,A: complex] :
      ( ( A4 = aa_complex_complex(plus_plus_complex(K2),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A4) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K2)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_7513_group__cancel_Oneg1,axiom,
    ! [A4: code_integer,K2: code_integer,A: code_integer] :
      ( ( A4 = plus_p5714425477246183910nteger(K2,A) )
     => ( uminus1351360451143612070nteger(A4) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(K2),uminus1351360451143612070nteger(A)) ) ) ).

% group_cancel.neg1
tff(fact_7514_group__cancel_Oneg1,axiom,
    ! [A4: rat,K2: rat,A: rat] :
      ( ( A4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K2)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_7515_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_7516_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% add.inverse_distrib_swap
tff(fact_7517_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_7518_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] : uminus1351360451143612070nteger(plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(B),uminus1351360451143612070nteger(A)) ).

% add.inverse_distrib_swap
tff(fact_7519_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% add.inverse_distrib_swap
tff(fact_7520_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_7521_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_7522_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_7523_one__neq__neg__one,axiom,
    one_one_Code_integer != uminus1351360451143612070nteger(one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_7524_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_7525_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_7526_minus__diff__commute,axiom,
    ! [B: real,A: real] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,B)),A) = aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,A)),B) ).

% minus_diff_commute
tff(fact_7527_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_7528_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(B),A) = minus_8373710615458151222nteger(uminus1351360451143612070nteger(A),B) ).

% minus_diff_commute
tff(fact_7529_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_diff_commute
tff(fact_7530_signed__take__bit__mult,axiom,
    ! [N: nat,K2: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),K2)),aa_int_int(bit_ri631733984087533419it_int(N),L))) = aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,K2),L)) ).

% signed_take_bit_mult
tff(fact_7531_fact__less__mono__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_7532_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).

% fact_ge_zero
tff(fact_7533_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).

% fact_ge_zero
tff(fact_7534_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_zero
tff(fact_7535_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).

% fact_ge_zero
tff(fact_7536_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,N)),zero_zero_rat)) ).

% fact_not_neg
tff(fact_7537_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,N)),zero_zero_int)) ).

% fact_not_neg
tff(fact_7538_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),zero_zero_nat)) ).

% fact_not_neg
tff(fact_7539_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,N)),zero_zero_real)) ).

% fact_not_neg
tff(fact_7540_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).

% fact_gt_zero
tff(fact_7541_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).

% fact_gt_zero
tff(fact_7542_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_gt_zero
tff(fact_7543_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).

% fact_gt_zero
tff(fact_7544_finite__set__decode,axiom,
    ! [N: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(nat_set_decode,N))) ).

% finite_set_decode
tff(fact_7545_fact__ge__1,axiom,
    ! [N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(semiri3624122377584611663nteger,N))) ).

% fact_ge_1
tff(fact_7546_fact__ge__1,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).

% fact_ge_1
tff(fact_7547_fact__ge__1,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).

% fact_ge_1
tff(fact_7548_fact__ge__1,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_1
tff(fact_7549_fact__ge__1,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).

% fact_ge_1
tff(fact_7550_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_num_real(numeral_numeral_real,N))) ).

% neg_numeral_le_numeral
tff(fact_7551_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M2))),numera6620942414471956472nteger(N))) ).

% neg_numeral_le_numeral
tff(fact_7552_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: 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,M2))),aa_num_rat(numeral_numeral_rat,N))) ).

% neg_numeral_le_numeral
tff(fact_7553_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: 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,M2))),aa_num_int(numeral_numeral_int,N))) ).

% neg_numeral_le_numeral
tff(fact_7554_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_7555_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M2)),uminus1351360451143612070nteger(numera6620942414471956472nteger(N)))) ).

% not_numeral_le_neg_numeral
tff(fact_7556_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_7557_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_7558_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% zero_neq_neg_numeral
tff(fact_7559_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) ).

% zero_neq_neg_numeral
tff(fact_7560_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% zero_neq_neg_numeral
tff(fact_7561_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_z3403309356797280102nteger != uminus1351360451143612070nteger(numera6620942414471956472nteger(N)) ).

% zero_neq_neg_numeral
tff(fact_7562_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).

% zero_neq_neg_numeral
tff(fact_7563_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_7564_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_7565_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M2)),uminus1351360451143612070nteger(numera6620942414471956472nteger(N)))) ).

% not_numeral_less_neg_numeral
tff(fact_7566_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_7567_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: 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,M2))),aa_num_int(numeral_numeral_int,N))) ).

% neg_numeral_less_numeral
tff(fact_7568_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_num_real(numeral_numeral_real,N))) ).

% neg_numeral_less_numeral
tff(fact_7569_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M2))),numera6620942414471956472nteger(N))) ).

% neg_numeral_less_numeral
tff(fact_7570_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: 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,M2))),aa_num_rat(numeral_numeral_rat,N))) ).

% neg_numeral_less_numeral
tff(fact_7571_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_7572_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer))) ).

% le_minus_one_simps(4)
tff(fact_7573_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_7574_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_7575_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_7576_le__minus__one__simps_I2_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),one_one_Code_integer)) ).

% le_minus_one_simps(2)
tff(fact_7577_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_7578_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_7579_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_7580_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = zero_zero_real )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_7581_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_7582_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
    <=> ( B = uminus1351360451143612070nteger(A) ) ) ).

% add_eq_0_iff
tff(fact_7583_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_7584_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_7585_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_7586_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_7587_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),A) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_7588_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_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_7589_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_7590_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B ) ) ).

% add.inverse_unique
tff(fact_7591_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_7592_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
     => ( uminus1351360451143612070nteger(A) = B ) ) ).

% add.inverse_unique
tff(fact_7593_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).

% add.inverse_unique
tff(fact_7594_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_7595_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( aa_real_real(plus_plus_real(A),B) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_7596_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_7597_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = uminus1351360451143612070nteger(B) )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_7598_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_7599_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_7600_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(plus_plus_real(A),B) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_7601_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_7602_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = B )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_7603_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_7604_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_7605_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_7606_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_7607_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != uminus1351360451143612070nteger(one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_7608_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_7609_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_7610_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_7611_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer))) ).

% less_minus_one_simps(4)
tff(fact_7612_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_7613_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_7614_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_7615_less__minus__one__simps_I2_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),one_one_Code_integer)) ).

% less_minus_one_simps(2)
tff(fact_7616_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_7617_numeral__times__minus__swap,axiom,
    ! [W2: num,X: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W2)),aa_int_int(uminus_uminus_int,X)) = aa_int_int(aa_int_fun_int_int(times_times_int,X),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W2))) ).

% numeral_times_minus_swap
tff(fact_7618_numeral__times__minus__swap,axiom,
    ! [W2: num,X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(uminus_uminus_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) ).

% numeral_times_minus_swap
tff(fact_7619_numeral__times__minus__swap,axiom,
    ! [W2: num,X: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(times_times_complex(X),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) ).

% numeral_times_minus_swap
tff(fact_7620_numeral__times__minus__swap,axiom,
    ! [W2: num,X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W2)),uminus1351360451143612070nteger(X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),uminus1351360451143612070nteger(numera6620942414471956472nteger(W2))) ).

% numeral_times_minus_swap
tff(fact_7621_numeral__times__minus__swap,axiom,
    ! [W2: num,X: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(uminus_uminus_rat,X)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) ).

% numeral_times_minus_swap
tff(fact_7622_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_7623_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_7624_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_7625_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_7626_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_7627_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_7628_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% one_neq_neg_numeral
tff(fact_7629_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N)) ).

% one_neq_neg_numeral
tff(fact_7630_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% one_neq_neg_numeral
tff(fact_7631_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_Code_integer != uminus1351360451143612070nteger(numera6620942414471956472nteger(N)) ).

% one_neq_neg_numeral
tff(fact_7632_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N)) ).

% one_neq_neg_numeral
tff(fact_7633_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,N) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_7634_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,N) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_7635_numeral__neq__neg__one,axiom,
    ! [N: num] : numera6690914467698888265omplex(N) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_7636_numeral__neq__neg__one,axiom,
    ! [N: num] : numera6620942414471956472nteger(N) != uminus1351360451143612070nteger(one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_7637_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,N) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% numeral_neq_neg_one
tff(fact_7638_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_7639_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_7640_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_7641_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 = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_7642_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(aa_rat_fun_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_7643_fact__mono,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,M2)),aa_nat_rat(semiri773545260158071498ct_rat,N))) ) ).

% fact_mono
tff(fact_7644_fact__mono,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,M2)),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ).

% fact_mono
tff(fact_7645_fact__mono,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).

% fact_mono
tff(fact_7646_fact__mono,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,M2)),aa_nat_real(semiri2265585572941072030t_real,N))) ) ).

% fact_mono
tff(fact_7647_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_7648_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),B) = aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_7649_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_7650_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,uminus1351360451143612070nteger(B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_7651_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_7652_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_7653_diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),B) = aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,B)) ).

% diff_conv_add_uminus
tff(fact_7654_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_7655_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,uminus1351360451143612070nteger(B)) ).

% diff_conv_add_uminus
tff(fact_7656_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% diff_conv_add_uminus
tff(fact_7657_group__cancel_Osub2,axiom,
    ! [B5: int,K2: int,B: int,A: int] :
      ( ( B5 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B5) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_7658_group__cancel_Osub2,axiom,
    ! [B5: real,K2: real,B: real,A: real] :
      ( ( B5 = aa_real_real(plus_plus_real(K2),B) )
     => ( aa_real_real(minus_minus_real(A),B5) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,K2)),aa_real_real(minus_minus_real(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_7659_group__cancel_Osub2,axiom,
    ! [B5: complex,K2: complex,B: complex,A: complex] :
      ( ( B5 = aa_complex_complex(plus_plus_complex(K2),B) )
     => ( aa_complex_complex(minus_minus_complex(A),B5) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K2)),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_7660_group__cancel_Osub2,axiom,
    ! [B5: code_integer,K2: code_integer,B: code_integer,A: code_integer] :
      ( ( B5 = plus_p5714425477246183910nteger(K2,B) )
     => ( minus_8373710615458151222nteger(A,B5) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(K2),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_7661_group__cancel_Osub2,axiom,
    ! [B5: rat,K2: rat,B: rat,A: rat] :
      ( ( B5 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B5) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_7662_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(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_neg_div
tff(fact_7663_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ( aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) ) ) ).

% dvd_neg_div
tff(fact_7664_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_7665_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(uminus1351360451143612070nteger(A),B) = uminus1351360451143612070nteger(divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_neg_div
tff(fact_7666_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(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) ) ) ).

% dvd_neg_div
tff(fact_7667_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(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_div_neg
tff(fact_7668_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ( aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) ) ) ).

% dvd_div_neg
tff(fact_7669_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_7670_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,uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_div_neg
tff(fact_7671_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(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) ) ) ).

% dvd_div_neg
tff(fact_7672_fact__dvd,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1406184849735516958ct_int,M2))) ) ).

% fact_dvd
tff(fact_7673_fact__dvd,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,M2))) ) ).

% fact_dvd
tff(fact_7674_fact__dvd,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,M2))) ) ).

% fact_dvd
tff(fact_7675_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_7676_int__of__nat__induct,axiom,
    ! [P2: fun_int_bool,Z2: int] :
      ( ! [N3: nat] : pp(aa_int_bool(P2,aa_nat_int(semiri1314217659103216013at_int,N3)))
     => ( ! [N3: nat] : pp(aa_int_bool(P2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3)))))
       => pp(aa_int_bool(P2,Z2)) ) ) ).

% int_of_nat_induct
tff(fact_7677_int__cases,axiom,
    ! [Z2: int] :
      ( ! [N3: nat] : Z2 != aa_nat_int(semiri1314217659103216013at_int,N3)
     => ~ ! [N3: nat] : Z2 != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).

% int_cases
tff(fact_7678_zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M2),N) = one_one_int )
    <=> ( ( ( M2 = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M2 = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( N = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_7679_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M2: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M2),N) = one_one_int )
     => ( ( M2 = one_one_int )
        | ( M2 = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_7680_zmod__zminus1__not__zero,axiom,
    ! [K2: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K2)),L) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
tff(fact_7681_zmod__zminus2__not__zero,axiom,
    ! [K2: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
tff(fact_7682_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),N) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(semiri5044797733671781792omplex,N)) ).

% pochhammer_same
tff(fact_7683_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(uminus1351360451143612070nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N))),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),N)),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ).

% pochhammer_same
tff(fact_7684_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).

% pochhammer_same
tff(fact_7685_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),N) = 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)),N)),aa_nat_int(semiri1406184849735516958ct_int,N)) ).

% pochhammer_same
tff(fact_7686_pochhammer__same,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(semiri2265585572941072030t_real,N)) ).

% pochhammer_same
tff(fact_7687_fact__ge__Suc__0__nat,axiom,
    ! [N: 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,N))) ).

% fact_ge_Suc_0_nat
tff(fact_7688_dvd__fact,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% dvd_fact
tff(fact_7689_not__zero__le__neg__numeral,axiom,
    ! [N: 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,N)))) ).

% not_zero_le_neg_numeral
tff(fact_7690_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(numera6620942414471956472nteger(N)))) ).

% not_zero_le_neg_numeral
tff(fact_7691_not__zero__le__neg__numeral,axiom,
    ! [N: 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,N)))) ).

% not_zero_le_neg_numeral
tff(fact_7692_not__zero__le__neg__numeral,axiom,
    ! [N: 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,N)))) ).

% not_zero_le_neg_numeral
tff(fact_7693_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))),zero_zero_real)) ).

% neg_numeral_le_zero
tff(fact_7694_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(N))),zero_z3403309356797280102nteger)) ).

% neg_numeral_le_zero
tff(fact_7695_neg__numeral__le__zero,axiom,
    ! [N: 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,N))),zero_zero_rat)) ).

% neg_numeral_le_zero
tff(fact_7696_neg__numeral__le__zero,axiom,
    ! [N: 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,N))),zero_zero_int)) ).

% neg_numeral_le_zero
tff(fact_7697_not__zero__less__neg__numeral,axiom,
    ! [N: 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,N)))) ).

% not_zero_less_neg_numeral
tff(fact_7698_not__zero__less__neg__numeral,axiom,
    ! [N: 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,N)))) ).

% not_zero_less_neg_numeral
tff(fact_7699_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(numera6620942414471956472nteger(N)))) ).

% not_zero_less_neg_numeral
tff(fact_7700_not__zero__less__neg__numeral,axiom,
    ! [N: 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,N)))) ).

% not_zero_less_neg_numeral
tff(fact_7701_neg__numeral__less__zero,axiom,
    ! [N: 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,N))),zero_zero_int)) ).

% neg_numeral_less_zero
tff(fact_7702_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))),zero_zero_real)) ).

% neg_numeral_less_zero
tff(fact_7703_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(N))),zero_z3403309356797280102nteger)) ).

% neg_numeral_less_zero
tff(fact_7704_neg__numeral__less__zero,axiom,
    ! [N: 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,N))),zero_zero_rat)) ).

% neg_numeral_less_zero
tff(fact_7705_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_7706_le__minus__one__simps_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),zero_z3403309356797280102nteger)) ).

% le_minus_one_simps(1)
tff(fact_7707_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_7708_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_7709_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_7710_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(one_one_Code_integer))) ).

% le_minus_one_simps(3)
tff(fact_7711_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_7712_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_7713_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_7714_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_7715_less__minus__one__simps_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),zero_z3403309356797280102nteger)) ).

% less_minus_one_simps(1)
tff(fact_7716_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_7717_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_7718_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_7719_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(one_one_Code_integer))) ).

% less_minus_one_simps(3)
tff(fact_7720_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_7721_not__one__le__neg__numeral,axiom,
    ! [M2: 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,M2)))) ).

% not_one_le_neg_numeral
tff(fact_7722_not__one__le__neg__numeral,axiom,
    ! [M2: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),uminus1351360451143612070nteger(numera6620942414471956472nteger(M2)))) ).

% not_one_le_neg_numeral
tff(fact_7723_not__one__le__neg__numeral,axiom,
    ! [M2: 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,M2)))) ).

% not_one_le_neg_numeral
tff(fact_7724_not__one__le__neg__numeral,axiom,
    ! [M2: 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,M2)))) ).

% not_one_le_neg_numeral
tff(fact_7725_not__numeral__le__neg__one,axiom,
    ! [M2: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_le_neg_one
tff(fact_7726_not__numeral__le__neg__one,axiom,
    ! [M2: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M2)),uminus1351360451143612070nteger(one_one_Code_integer))) ).

% not_numeral_le_neg_one
tff(fact_7727_not__numeral__le__neg__one,axiom,
    ! [M2: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% not_numeral_le_neg_one
tff(fact_7728_not__numeral__le__neg__one,axiom,
    ! [M2: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(uminus_uminus_int,one_one_int))) ).

% not_numeral_le_neg_one
tff(fact_7729_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),aa_real_real(uminus_uminus_real,one_one_real))) ).

% neg_numeral_le_neg_one
tff(fact_7730_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M2))),uminus1351360451143612070nteger(one_one_Code_integer))) ).

% neg_numeral_le_neg_one
tff(fact_7731_neg__numeral__le__neg__one,axiom,
    ! [M2: 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,M2))),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% neg_numeral_le_neg_one
tff(fact_7732_neg__numeral__le__neg__one,axiom,
    ! [M2: 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,M2))),aa_int_int(uminus_uminus_int,one_one_int))) ).

% neg_numeral_le_neg_one
tff(fact_7733_neg__one__le__numeral,axiom,
    ! [M2: 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,M2))) ).

% neg_one_le_numeral
tff(fact_7734_neg__one__le__numeral,axiom,
    ! [M2: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),numera6620942414471956472nteger(M2))) ).

% neg_one_le_numeral
tff(fact_7735_neg__one__le__numeral,axiom,
    ! [M2: 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,M2))) ).

% neg_one_le_numeral
tff(fact_7736_neg__one__le__numeral,axiom,
    ! [M2: 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,M2))) ).

% neg_one_le_numeral
tff(fact_7737_neg__numeral__le__one,axiom,
    ! [M2: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),one_one_real)) ).

% neg_numeral_le_one
tff(fact_7738_neg__numeral__le__one,axiom,
    ! [M2: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M2))),one_one_Code_integer)) ).

% neg_numeral_le_one
tff(fact_7739_neg__numeral__le__one,axiom,
    ! [M2: 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,M2))),one_one_rat)) ).

% neg_numeral_le_one
tff(fact_7740_neg__numeral__le__one,axiom,
    ! [M2: 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,M2))),one_one_int)) ).

% neg_numeral_le_one
tff(fact_7741_gbinomial__pochhammer,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),K2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),K2))),aa_nat_complex(semiri5044797733671781792omplex,K2)) ).

% gbinomial_pochhammer
tff(fact_7742_gbinomial__pochhammer,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,A)),K2))),aa_nat_rat(semiri773545260158071498ct_rat,K2)) ).

% gbinomial_pochhammer
tff(fact_7743_gbinomial__pochhammer,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,A)),K2))),aa_nat_real(semiri2265585572941072030t_real,K2)) ).

% gbinomial_pochhammer
tff(fact_7744_neg__numeral__less__one,axiom,
    ! [M2: 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,M2))),one_one_int)) ).

% neg_numeral_less_one
tff(fact_7745_neg__numeral__less__one,axiom,
    ! [M2: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M2))),one_one_real)) ).

% neg_numeral_less_one
tff(fact_7746_neg__numeral__less__one,axiom,
    ! [M2: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M2))),one_one_Code_integer)) ).

% neg_numeral_less_one
tff(fact_7747_neg__numeral__less__one,axiom,
    ! [M2: 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,M2))),one_one_rat)) ).

% neg_numeral_less_one
tff(fact_7748_neg__one__less__numeral,axiom,
    ! [M2: 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,M2))) ).

% neg_one_less_numeral
tff(fact_7749_neg__one__less__numeral,axiom,
    ! [M2: 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,M2))) ).

% neg_one_less_numeral
tff(fact_7750_neg__one__less__numeral,axiom,
    ! [M2: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),numera6620942414471956472nteger(M2))) ).

% neg_one_less_numeral
tff(fact_7751_neg__one__less__numeral,axiom,
    ! [M2: 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,M2))) ).

% neg_one_less_numeral
tff(fact_7752_not__numeral__less__neg__one,axiom,
    ! [M2: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(uminus_uminus_int,one_one_int))) ).

% not_numeral_less_neg_one
tff(fact_7753_not__numeral__less__neg__one,axiom,
    ! [M2: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M2)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_less_neg_one
tff(fact_7754_not__numeral__less__neg__one,axiom,
    ! [M2: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M2)),uminus1351360451143612070nteger(one_one_Code_integer))) ).

% not_numeral_less_neg_one
tff(fact_7755_not__numeral__less__neg__one,axiom,
    ! [M2: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M2)),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% not_numeral_less_neg_one
tff(fact_7756_not__one__less__neg__numeral,axiom,
    ! [M2: 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,M2)))) ).

% not_one_less_neg_numeral
tff(fact_7757_not__one__less__neg__numeral,axiom,
    ! [M2: 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,M2)))) ).

% not_one_less_neg_numeral
tff(fact_7758_not__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),uminus1351360451143612070nteger(numera6620942414471956472nteger(M2)))) ).

% not_one_less_neg_numeral
tff(fact_7759_not__one__less__neg__numeral,axiom,
    ! [M2: 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,M2)))) ).

% not_one_less_neg_numeral
tff(fact_7760_not__neg__one__less__neg__numeral,axiom,
    ! [M2: 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,M2)))) ).

% not_neg_one_less_neg_numeral
tff(fact_7761_not__neg__one__less__neg__numeral,axiom,
    ! [M2: 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,M2)))) ).

% not_neg_one_less_neg_numeral
tff(fact_7762_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),uminus1351360451143612070nteger(numera6620942414471956472nteger(M2)))) ).

% not_neg_one_less_neg_numeral
tff(fact_7763_not__neg__one__less__neg__numeral,axiom,
    ! [M2: 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,M2)))) ).

% not_neg_one_less_neg_numeral
tff(fact_7764_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C2)) )
    <=> ( ( ( C2 != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = aa_real_real(uminus_uminus_real,B) ) )
        & ( ( C2 = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_7765_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C2: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C2)) )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),C2) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
        & ( ( C2 = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_7766_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)) )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = aa_rat_rat(uminus_uminus_rat,B) ) )
        & ( ( C2 = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_7767_minus__divide__eq__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C2)) = A )
    <=> ( ( ( C2 != zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) ) )
        & ( ( C2 = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_7768_minus__divide__eq__eq,axiom,
    ! [B: complex,C2: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C2)) = A )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(times_times_complex(A),C2) ) )
        & ( ( C2 = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_7769_minus__divide__eq__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)) = A )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) ) )
        & ( ( C2 = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_7770_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) = C2 )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_7771_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C2: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = C2 )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(times_times_complex(C2),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_7772_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) = C2 )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_7773_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C2 = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_7774_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C2: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C2 = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) )
      <=> ( aa_complex_complex(times_times_complex(C2),B) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_7775_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C2 = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_7776_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_7777_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_7778_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_7779_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(one))),B) = uminus1351360451143612070nteger(B) ).

% mult_1s_ring_1(1)
tff(fact_7780_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(aa_rat_fun_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_7781_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_7782_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_7783_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_7784_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),uminus1351360451143612070nteger(numera6620942414471956472nteger(one))) = uminus1351360451143612070nteger(B) ).

% mult_1s_ring_1(2)
tff(fact_7785_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(aa_rat_fun_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_7786_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),B) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B != zero_zero_real )
        & ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_7787_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_7788_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_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_7789_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_7790_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_7791_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_7792_uminus__numeral__One,axiom,
    uminus1351360451143612070nteger(numera6620942414471956472nteger(one)) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_7793_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_7794_fact__less__mono,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,M2)),aa_nat_rat(semiri773545260158071498ct_rat,N))) ) ) ).

% fact_less_mono
tff(fact_7795_fact__less__mono,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,M2)),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ) ).

% fact_less_mono
tff(fact_7796_fact__less__mono,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% fact_less_mono
tff(fact_7797_fact__less__mono,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,M2)),aa_nat_real(semiri2265585572941072030t_real,N))) ) ) ).

% fact_less_mono
tff(fact_7798_power__minus,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = 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)),N)),aa_nat_int(power_power_int(A),N)) ).

% power_minus
tff(fact_7799_power__minus,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% power_minus
tff(fact_7800_power__minus,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(power_power_complex(A),N)) ).

% power_minus
tff(fact_7801_power__minus,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).

% power_minus
tff(fact_7802_power__minus,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(A),N)) ).

% power_minus
tff(fact_7803_inf__shunt,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = bot_bo2099793752762293965at_nat )
    <=> ord_le3146513528884898305at_nat(X,uminus6524753893492686040at_nat(Y)) ) ).

% inf_shunt
tff(fact_7804_inf__shunt,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( inf_inf_set_int(X,Y) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X),uminus1532241313380277803et_int(Y))) ) ).

% inf_shunt
tff(fact_7805_inf__shunt,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( inf_inf_set_real(X,Y) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(X),uminus612125837232591019t_real(Y))) ) ).

% inf_shunt
tff(fact_7806_inf__shunt,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),uminus5710092332889474511et_nat(Y))) ) ).

% inf_shunt
tff(fact_7807_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,N))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N)))) ).

% fact_fact_dvd_fact
tff(fact_7808_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N: 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,K2)),aa_nat_int(semiri1406184849735516958ct_int,N))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N)))) ).

% fact_fact_dvd_fact
tff(fact_7809_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N: 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,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,N))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N)))) ).

% fact_fact_dvd_fact
tff(fact_7810_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N: 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,K2)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N)))) ).

% fact_fact_dvd_fact
tff(fact_7811_fact__mod,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1406184849735516958ct_int,M2)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_7812_fact__mod,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( modulo8411746178871703098atural(semiri2447717529341329178atural(N),semiri2447717529341329178atural(M2)) = zero_z2226904508553997617atural ) ) ).

% fact_mod
tff(fact_7813_fact__mod,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,M2)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_7814_sup__neg__inf,axiom,
    ! [P: set_Pr1261947904930325089at_nat,Q2: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(P,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Q2),R))
    <=> ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,P),uminus6524753893492686040at_nat(Q2)),R) ) ).

% sup_neg_inf
tff(fact_7815_sup__neg__inf,axiom,
    ! [P: set_nat,Q2: set_nat,R: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(P),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Q2),R)))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,P),uminus5710092332889474511et_nat(Q2))),R)) ) ).

% sup_neg_inf
tff(fact_7816_shunt2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),uminus6524753893492686040at_nat(Y)),Z2)
    <=> ord_le3146513528884898305at_nat(X,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z2)) ) ).

% shunt2
tff(fact_7817_shunt2,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),uminus5710092332889474511et_nat(Y))),Z2))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z2))) ) ).

% shunt2
tff(fact_7818_shunt1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y),Z2)
    <=> ord_le3146513528884898305at_nat(X,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat(Y)),Z2)) ) ).

% shunt1
tff(fact_7819_shunt1,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Z2))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(Y)),Z2))) ) ).

% shunt1
tff(fact_7820_fact__le__power,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,N)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_7821_fact__le__power,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_7822_fact__le__power,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_7823_fact__le__power,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_7824_int__cases4,axiom,
    ! [M2: int] :
      ( ! [N3: nat] : M2 != aa_nat_int(semiri1314217659103216013at_int,N3)
     => ~ ! [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
           => ( M2 != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) ) ) ) ).

% int_cases4
tff(fact_7825_int__zle__neg,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M2))))
    <=> ( ( N = zero_zero_nat )
        & ( M2 = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_7826_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = zero_zero_int ) )
      & ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ) ).

% zmod_zminus1_eq_if
tff(fact_7827_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = zero_zero_int ) )
      & ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) != zero_zero_int )
       => ( aa_int_int(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ) ) ).

% zmod_zminus2_eq_if
tff(fact_7828_fact__div__fact__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R),N))
     => 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,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),R)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R))) ) ).

% fact_div_fact_le_pow
tff(fact_7829_binomial__fact__lemma,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_7830_subset__decode__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(nat_set_decode,M2)),aa_nat_set_nat(nat_set_decode,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% subset_decode_imp_le
tff(fact_7831_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C2))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_7832_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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(aa_rat_fun_rat_rat(times_times_rat,A),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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_7833_minus__divide__less__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C2))),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => 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),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_7834_minus__divide__less__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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_7835_neg__less__minus__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C2))))
      <=> 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),C2))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_7836_neg__less__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_7837_neg__minus__divide__less__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C2))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_7838_neg__minus__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_7839_pos__less__minus__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C2))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_7840_pos__less__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_7841_pos__minus__divide__less__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C2))),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),C2))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_7842_pos__minus__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_7843_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = aa_real_real(divide_divide_real(B),C2) )
    <=> ( ( ( C2 != 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))),C2) = B ) )
        & ( ( C2 = 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_7844_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: complex,C2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = aa_complex_complex(divide1717551699836669952omplex(B),C2) )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C2) = B ) )
        & ( ( C2 = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_7845_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2) )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2) = B ) )
        & ( ( C2 = 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_7846_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( ( aa_real_real(divide_divide_real(B),C2) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) )
    <=> ( ( ( C2 != 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))),C2) ) )
        & ( ( C2 = 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_7847_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C2: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C2) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) )
    <=> ( ( ( C2 != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C2) ) )
        & ( ( C2 = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_7848_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) )
    <=> ( ( ( C2 != zero_zero_rat )
         => ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2) ) )
        & ( ( C2 = 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_7849_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z2))),B) = B ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z2))),B) = aa_real_real(divide_divide_real(aa_real_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_7850_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_7851_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z2))),B) = B ) )
      & ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z2))),B) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_7852_minus__divide__add__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(X),Z2))),Y) = aa_real_real(divide_divide_real(aa_real_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_7853_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_7854_minus__divide__add__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z2))),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z2))),Z2) ) ) ).

% minus_divide_add_eq_iff
tff(fact_7855_signed__take__bit__int__greater__eq,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))),aa_int_int(bit_ri631733984087533419it_int(N),K2))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_7856_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z2))),B) = aa_real_real(uminus_uminus_real,B) ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z2))),B) = aa_real_real(divide_divide_real(aa_real_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_7857_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_7858_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z2))),B) = aa_rat_rat(uminus_uminus_rat,B) ) )
      & ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z2))),B) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_7859_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A),Z2)),B) = aa_real_real(uminus_uminus_real,B) ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A),Z2)),B) = aa_real_real(divide_divide_real(aa_real_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_7860_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_7861_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z2)),B) = aa_rat_rat(uminus_uminus_rat,B) ) )
      & ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z2)),B) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_7862_minus__divide__diff__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(X),Z2))),Y) = aa_real_real(divide_divide_real(aa_real_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_7863_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_7864_minus__divide__diff__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z2))),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z2))),Z2) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_7865_choose__dvd,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))),aa_nat_rat(semiri773545260158071498ct_rat,N))) ) ).

% choose_dvd
tff(fact_7866_choose__dvd,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => 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,K2)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ).

% choose_dvd
tff(fact_7867_choose__dvd,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => 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,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).

% choose_dvd
tff(fact_7868_choose__dvd,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))),aa_nat_real(semiri2265585572941072030t_real,N))) ) ).

% choose_dvd
tff(fact_7869_int__cases3,axiom,
    ! [K2: int] :
      ( ( K2 != zero_zero_int )
     => ( ! [N3: nat] :
            ( ( K2 = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3)) )
       => ~ ! [N3: nat] :
              ( ( K2 = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3)) ) ) ) ).

% int_cases3
tff(fact_7870_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K2) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7871_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(uminus1351360451143612070nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K2) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7872_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K2) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7873_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K2) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7874_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K2) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7875_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K2) = zero_zero_complex )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7876_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(uminus1351360451143612070nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K2) = zero_z3403309356797280102nteger )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7877_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K2) = zero_zero_rat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7878_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K2) = zero_zero_real )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7879_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K2) = zero_zero_int )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7880_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = zero_zero_complex )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7881_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7882_pochhammer__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = zero_zero_real )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7883_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K2) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7884_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(uminus1351360451143612070nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K2) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7885_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K2) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7886_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K2) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7887_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K2) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7888_not__zle__0__negative,axiom,
    ! [N: 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,N))))) ).

% not_zle_0_negative
tff(fact_7889_negD,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
     => ? [N3: nat] : X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).

% negD
tff(fact_7890_negative__zless__0,axiom,
    ! [N: 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,N)))),zero_zero_int)) ).

% negative_zless_0
tff(fact_7891_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(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_7892_prod_OIf__cases,axiom,
    ! [A4: set_real,P2: fun_real_bool,H: fun_real_real,G2: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_tx(P2),H),G2),A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups1681761925125756287l_real(H,inf_inf_set_real(A4,collect_real(P2)))),groups1681761925125756287l_real(G2,inf_inf_set_real(A4,uminus612125837232591019t_real(collect_real(P2))))) ) ) ).

% prod.If_cases
tff(fact_7893_prod_OIf__cases,axiom,
    ! [A4: set_int,P2: fun_int_bool,H: fun_int_real,G2: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_ty(P2),H),G2),A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups2316167850115554303t_real(H,inf_inf_set_int(A4,collect_int(P2)))),groups2316167850115554303t_real(G2,inf_inf_set_int(A4,uminus1532241313380277803et_int(collect_int(P2))))) ) ) ).

% prod.If_cases
tff(fact_7894_prod_OIf__cases,axiom,
    ! [A4: set_complex,P2: fun_complex_bool,H: fun_complex_real,G2: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_tz(P2),H),G2),A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(H,inf_inf_set_complex(A4,collect_complex(P2)))),groups766887009212190081x_real(G2,inf_inf_set_complex(A4,uminus8566677241136511917omplex(collect_complex(P2))))) ) ) ).

% prod.If_cases
tff(fact_7895_prod_OIf__cases,axiom,
    ! [A4: set_nat,P2: fun_nat_bool,H: fun_nat_real,G2: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_ua(P2),H),G2)),A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(H),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(P2)))),aa_set_nat_real(groups129246275422532515t_real(G2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),uminus5710092332889474511et_nat(collect_nat(P2))))) ) ) ).

% prod.If_cases
tff(fact_7896_prod_OIf__cases,axiom,
    ! [A4: set_real,P2: fun_real_bool,H: fun_real_rat,G2: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_ub(P2),H),G2),A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups4061424788464935467al_rat(H,inf_inf_set_real(A4,collect_real(P2)))),groups4061424788464935467al_rat(G2,inf_inf_set_real(A4,uminus612125837232591019t_real(collect_real(P2))))) ) ) ).

% prod.If_cases
tff(fact_7897_prod_OIf__cases,axiom,
    ! [A4: set_int,P2: fun_int_bool,H: fun_int_rat,G2: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_uc(P2),H),G2),A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(H,inf_inf_set_int(A4,collect_int(P2)))),groups1072433553688619179nt_rat(G2,inf_inf_set_int(A4,uminus1532241313380277803et_int(collect_int(P2))))) ) ) ).

% prod.If_cases
tff(fact_7898_prod_OIf__cases,axiom,
    ! [A4: set_complex,P2: fun_complex_bool,H: fun_complex_rat,G2: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A4))
     => ( groups225925009352817453ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_ud(P2),H),G2),A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups225925009352817453ex_rat(H,inf_inf_set_complex(A4,collect_complex(P2)))),groups225925009352817453ex_rat(G2,inf_inf_set_complex(A4,uminus8566677241136511917omplex(collect_complex(P2))))) ) ) ).

% prod.If_cases
tff(fact_7899_prod_OIf__cases,axiom,
    ! [A4: set_nat,P2: fun_nat_bool,H: fun_nat_rat,G2: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ue(P2),H),G2),A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(H,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(P2)))),groups73079841787564623at_rat(G2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),uminus5710092332889474511et_nat(collect_nat(P2))))) ) ) ).

% prod.If_cases
tff(fact_7900_prod_OIf__cases,axiom,
    ! [A4: set_real,P2: fun_real_bool,H: fun_real_nat,G2: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A4))
     => ( groups4696554848551431203al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_uf(P2),H),G2),A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups4696554848551431203al_nat(H,inf_inf_set_real(A4,collect_real(P2)))),groups4696554848551431203al_nat(G2,inf_inf_set_real(A4,uminus612125837232591019t_real(collect_real(P2))))) ) ) ).

% prod.If_cases
tff(fact_7901_prod_OIf__cases,axiom,
    ! [A4: set_int,P2: fun_int_bool,H: fun_int_nat,G2: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A4))
     => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_ug(P2),H),G2),A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(H,inf_inf_set_int(A4,collect_int(P2)))),groups1707563613775114915nt_nat(G2,inf_inf_set_int(A4,uminus1532241313380277803et_int(collect_int(P2))))) ) ) ).

% prod.If_cases
tff(fact_7902_binomial__altdef__nat,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))) ) ) ).

% binomial_altdef_nat
tff(fact_7903_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C2))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_7904_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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(aa_rat_fun_rat_rat(times_times_rat,A),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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_7905_minus__divide__le__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C2))),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => 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),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_7906_minus__divide__le__eq,axiom,
    ! [B: rat,C2: 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(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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_7907_neg__le__minus__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C2))))
      <=> 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),C2))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_7908_neg__le__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_7909_neg__minus__divide__le__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C2))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_7910_neg__minus__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_7911_pos__le__minus__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C2))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_7912_pos__le__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_7913_pos__minus__divide__le__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C2))),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),C2))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_7914_pos__minus__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_7915_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(divide_divide_real(B),C2)),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),C2))
         => 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))),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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))),C2)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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_7916_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),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),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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_7917_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C2: 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(divide_divide_real(B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => 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))),C2)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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))),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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_7918_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C2: 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(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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_7919_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( 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,N),K2)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7920_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K2)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7921_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K2)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7922_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K2)) = aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7923_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( 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,N),K2)) = 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,N),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7924_neg__int__cases,axiom,
    ! [K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
     => ~ ! [N3: nat] :
            ( ( K2 = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3)) ) ) ).

% neg_int_cases
tff(fact_7925_minus__mod__int__eq,axiom,
    ! [L: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K2)),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_7926_zmod__minus1,axiom,
    ! [B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),one_one_int) ) ) ).

% zmod_minus1
tff(fact_7927_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) )
        & ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) != zero_zero_int )
         => ( aa_int_int(aa_int_fun_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(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int) ) ) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_7928_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) )
        & ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) != zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int) ) ) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_7929_fact__eq__fact__times,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M2))) ) ) ).

% fact_eq_fact_times
tff(fact_7930_signed__take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( N = zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N,A) = uminus1351360451143612070nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) )
      & ( ( N != zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N,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,N),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_7931_signed__take__bit__rec,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_int_int(bit_ri631733984087533419it_int(N),A) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
      & ( ( N != zero_zero_nat )
       => ( aa_int_int(bit_ri631733984087533419it_int(N),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_7932_square__fact__le__2__fact,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,N))),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))),N)))) ).

% square_fact_le_2_fact
tff(fact_7933_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),product_Pair_int_int(Q2,R)))
     => ( ( B != zero_zero_int )
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R),zero_zero_int)),aa_int_int(uminus_uminus_int,Q2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q2)),one_one_int)),aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R),zero_zero_int)),zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R))))) ) ) ).

% zminus1_lemma
tff(fact_7934_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(B),C2)),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),C2))
         => 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))),C2))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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))),C2)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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_7935_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),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),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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_7936_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C2: 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(divide_divide_real(B),C2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => 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))),C2)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C2))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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))),C2))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),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_7937_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C2: 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(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C2))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C2))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C2),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_7938_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(aa_real_fun_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_7939_square__le__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,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_7940_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_7941_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_7942_minus__power__mult__self,axiom,
    ! [A: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N)) = 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))),N)) ).

% minus_power_mult_self
tff(fact_7943_minus__power__mult__self,axiom,
    ! [A: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_7944_minus__power__mult__self,axiom,
    ! [A: complex,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_7945_minus__power__mult__self,axiom,
    ! [A: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_7946_minus__power__mult__self,axiom,
    ! [A: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N)) = 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))),N)) ).

% minus_power_mult_self
tff(fact_7947_fact__num__eq__if,axiom,
    ! [M2: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_Code_integer(semiri3624122377584611663nteger,M2) = one_one_Code_integer ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_Code_integer(semiri3624122377584611663nteger,M2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M2)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7948_fact__num__eq__if,axiom,
    ! [M2: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_complex(semiri5044797733671781792omplex,M2) = one_one_complex ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_complex(semiri5044797733671781792omplex,M2) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M2)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7949_fact__num__eq__if,axiom,
    ! [M2: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_rat(semiri773545260158071498ct_rat,M2) = one_one_rat ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_rat(semiri773545260158071498ct_rat,M2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7950_fact__num__eq__if,axiom,
    ! [M2: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_int(semiri1406184849735516958ct_int,M2) = one_one_int ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_int(semiri1406184849735516958ct_int,M2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7951_fact__num__eq__if,axiom,
    ! [M2: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_nat(semiri1408675320244567234ct_nat,M2) = one_one_nat ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_nat(semiri1408675320244567234ct_nat,M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7952_fact__num__eq__if,axiom,
    ! [M2: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_real(semiri2265585572941072030t_real,M2) = one_one_real ) )
      & ( ( M2 != zero_zero_nat )
       => ( aa_nat_real(semiri2265585572941072030t_real,M2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7953_fact__code,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) = 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))),N),one_one_nat)) ).

% fact_code
tff(fact_7954_fact__code,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) = 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))),N),one_one_nat)) ).

% fact_code
tff(fact_7955_fact__code,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,N) = 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))),N),one_one_nat)) ).

% fact_code
tff(fact_7956_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7957_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(semiri1406184849735516958ct_int,N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7958_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7959_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_real(semiri2265585572941072030t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7960_pochhammer__absorb__comp,axiom,
    ! [R: complex,K2: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R),aa_nat_complex(semiri8010041392384452111omplex,K2))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R)),K2)) = aa_complex_complex(times_times_complex(R),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R)),one_one_complex)),K2)) ).

% pochhammer_absorb_comp
tff(fact_7961_pochhammer__absorb__comp,axiom,
    ! [R: code_integer,K2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(R,aa_nat_Code_integer(semiri4939895301339042750nteger,K2))),aa_nat_Code_integer(comm_s8582702949713902594nteger(uminus1351360451143612070nteger(R)),K2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(uminus1351360451143612070nteger(R),one_one_Code_integer)),K2)) ).

% pochhammer_absorb_comp
tff(fact_7962_pochhammer__absorb__comp,axiom,
    ! [R: rat,K2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,R),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R)),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,R)),one_one_rat)),K2)) ).

% pochhammer_absorb_comp
tff(fact_7963_pochhammer__absorb__comp,axiom,
    ! [R: real,K2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(R),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,R)),K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,R)),one_one_real)),K2)) ).

% pochhammer_absorb_comp
tff(fact_7964_pochhammer__absorb__comp,axiom,
    ! [R: int,K2: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R),aa_nat_int(semiri1314217659103216013at_int,K2))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R)),K2)) = aa_int_int(aa_int_fun_int_int(times_times_int,R),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R)),one_one_int)),K2)) ).

% pochhammer_absorb_comp
tff(fact_7965_gbinomial__index__swap,axiom,
    ! [K2: nat,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),one_one_complex)),K2)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),N)) ).

% gbinomial_index_swap
tff(fact_7966_gbinomial__index__swap,axiom,
    ! [K2: nat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),one_one_rat)),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),N)) ).

% gbinomial_index_swap
tff(fact_7967_gbinomial__index__swap,axiom,
    ! [K2: nat,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),one_one_real)),K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),N)) ).

% gbinomial_index_swap
tff(fact_7968_gbinomial__negated__upper,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,K2)),A)),one_one_complex)),K2)) ).

% gbinomial_negated_upper
tff(fact_7969_gbinomial__negated__upper,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)),A)),one_one_rat)),K2)) ).

% gbinomial_negated_upper
tff(fact_7970_gbinomial__negated__upper,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,K2)),A)),one_one_real)),K2)) ).

% gbinomial_negated_upper
tff(fact_7971_binomial__fact,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,N)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K2)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))) ) ) ).

% binomial_fact
tff(fact_7972_binomial__fact,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri773545260158071498ct_rat,N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))) ) ) ).

% binomial_fact
tff(fact_7973_binomial__fact,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2)) = aa_real_real(divide_divide_real(aa_nat_real(semiri2265585572941072030t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)))) ) ) ).

% binomial_fact
tff(fact_7974_fact__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K2)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2))) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,N)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2))) ) ) ).

% fact_binomial
tff(fact_7975_fact__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri773545260158071498ct_rat,N)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2))) ) ) ).

% fact_binomial
tff(fact_7976_fact__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2))) = aa_real_real(divide_divide_real(aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2))) ) ) ).

% fact_binomial
tff(fact_7977_Bernoulli__inequality,axiom,
    ! [X: real,N: 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(plus_plus_real(one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(one_one_real),X)),N))) ) ).

% Bernoulli_inequality
tff(fact_7978_fact__div__fact,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M2)) ) ) ).

% fact_div_fact
tff(fact_7979_power__minus1__odd,axiom,
    ! [N: 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))),N))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_7980_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_7981_power__minus1__odd,axiom,
    ! [N: 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))),N))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_7982_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(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))),N))) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% power_minus1_odd
tff(fact_7983_power__minus1__odd,axiom,
    ! [N: 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))),N))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% power_minus1_odd
tff(fact_7984_gbinomial__minus,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2)) ).

% gbinomial_minus
tff(fact_7985_gbinomial__minus,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A)),K2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2)) ).

% gbinomial_minus
tff(fact_7986_gbinomial__minus,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(uminus_uminus_real,A)),K2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2)) ).

% gbinomial_minus
tff(fact_7987_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K2: 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))),N)),K2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(N),K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))))) ) ).

% signed_take_bit_int_less_eq
tff(fact_7988_pochhammer__minus_H,axiom,
    ! [B: complex,K2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K2)) ).

% pochhammer_minus'
tff(fact_7989_pochhammer__minus_H,axiom,
    ! [B: code_integer,K2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K2)),one_one_Code_integer)),K2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),K2)),aa_nat_Code_integer(comm_s8582702949713902594nteger(uminus1351360451143612070nteger(B)),K2)) ).

% pochhammer_minus'
tff(fact_7990_pochhammer__minus_H,axiom,
    ! [B: rat,K2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K2)) ).

% pochhammer_minus'
tff(fact_7991_pochhammer__minus_H,axiom,
    ! [B: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(B),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K2)) ).

% pochhammer_minus'
tff(fact_7992_pochhammer__minus_H,axiom,
    ! [B: int,K2: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K2))),one_one_int)),K2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K2)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K2)) ).

% pochhammer_minus'
tff(fact_7993_pochhammer__minus,axiom,
    ! [B: complex,K2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2)) ).

% pochhammer_minus
tff(fact_7994_pochhammer__minus,axiom,
    ! [B: code_integer,K2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(uminus1351360451143612070nteger(B)),K2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),K2)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K2)),one_one_Code_integer)),K2)) ).

% pochhammer_minus
tff(fact_7995_pochhammer__minus,axiom,
    ! [B: rat,K2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2)) ).

% pochhammer_minus
tff(fact_7996_pochhammer__minus,axiom,
    ! [B: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(B),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2)) ).

% pochhammer_minus
tff(fact_7997_pochhammer__minus,axiom,
    ! [B: int,K2: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K2)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K2))),one_one_int)),K2)) ).

% pochhammer_minus
tff(fact_7998_int__bit__induct,axiom,
    ! [P2: fun_int_bool,K2: int] :
      ( pp(aa_int_bool(P2,zero_zero_int))
     => ( pp(aa_int_bool(P2,aa_int_int(uminus_uminus_int,one_one_int)))
       => ( ! [K: int] :
              ( pp(aa_int_bool(P2,K))
             => ( ( K != zero_zero_int )
               => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
         => ( ! [K: int] :
                ( pp(aa_int_bool(P2,K))
               => ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
                 => pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => pp(aa_int_bool(P2,K2)) ) ) ) ) ).

% int_bit_induct
tff(fact_7999_gbinomial__sum__lower__neg,axiom,
    ! [A: complex,M2: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_uh(A)),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),M2)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),M2)) ).

% gbinomial_sum_lower_neg
tff(fact_8000_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,M2: nat] : groups2906978787729119204at_rat(aTP_Lamm_ui(A),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),M2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),M2)) ).

% gbinomial_sum_lower_neg
tff(fact_8001_gbinomial__sum__lower__neg,axiom,
    ! [A: real,M2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_uj,A)),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),M2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),M2)) ).

% gbinomial_sum_lower_neg
tff(fact_8002_gbinomial__Suc,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2)) = aa_complex_complex(divide1717551699836669952omplex(aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_uk(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2))),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_8003_gbinomial__Suc,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,groups73079841787564623at_rat(aTP_Lamm_ul(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_8004_gbinomial__Suc,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2)) = aa_real_real(divide_divide_real(aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_um,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_8005_gbinomial__Suc,axiom,
    ! [A: int,K2: nat] : aa_nat_int(gbinomial_int(A),aa_nat_nat(suc,K2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,groups705719431365010083at_int(aTP_Lamm_un(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_8006_gbinomial__Suc,axiom,
    ! [A: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uo,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_8007_gbinomial__partial__sum__poly,axiom,
    ! [M2: nat,A: complex,X: complex,Y: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_qc(M2),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_up(M2),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M2)) ).

% gbinomial_partial_sum_poly
tff(fact_8008_gbinomial__partial__sum__poly,axiom,
    ! [M2: nat,A: rat,X: rat,Y: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_qe(M2),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M2)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_uq(M2),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M2)) ).

% gbinomial_partial_sum_poly
tff(fact_8009_gbinomial__partial__sum__poly,axiom,
    ! [M2: 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_qg(M2),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_ur(M2),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M2)) ).

% gbinomial_partial_sum_poly
tff(fact_8010_root__polyfun,axiom,
    ! [N: nat,Z2: int,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_int(power_power_int(Z2),N) = A )
      <=> ( groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_us(N),Z2),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ) ).

% root_polyfun
tff(fact_8011_root__polyfun,axiom,
    ! [N: nat,Z2: complex,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_complex(power_power_complex(Z2),N) = A )
      <=> ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_ut(N),Z2),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex ) ) ) ).

% root_polyfun
tff(fact_8012_root__polyfun,axiom,
    ! [N: nat,Z2: code_integer,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Z2),N) = A )
      <=> ( groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_uu(N),Z2),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
tff(fact_8013_root__polyfun,axiom,
    ! [N: nat,Z2: rat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_rat(power_power_rat(Z2),N) = A )
      <=> ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_uv(N),Z2),A),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat ) ) ) ).

% root_polyfun
tff(fact_8014_root__polyfun,axiom,
    ! [N: nat,Z2: real,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),N) = A )
      <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_uw(N),Z2),A)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real ) ) ) ).

% root_polyfun
tff(fact_8015_set__decode__plus__power__2,axiom,
    ! [N: nat,Z2: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(N),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))),N)),Z2)) = insert_nat(N,aa_nat_set_nat(nat_set_decode,Z2)) ) ) ).

% set_decode_plus_power_2
tff(fact_8016_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_ux(N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
tff(fact_8017_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups7501900531339628137nteger(aTP_Lamm_uy(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
tff(fact_8018_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups2906978787729119204at_rat(aTP_Lamm_uz(N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_8019_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_va,N),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_8020_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_vb,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = zero_zero_real ) ) ).

% choose_alternating_linear_sum
tff(fact_8021_set__decode__def,axiom,
    ! [X: nat] : aa_nat_set_nat(nat_set_decode,X) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_vc,X)) ).

% set_decode_def
tff(fact_8022_fact__double,axiom,
    ! [N: 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))),N)) = 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))),N))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one)))),N))),aa_nat_complex(semiri5044797733671781792omplex,N)) ).

% fact_double
tff(fact_8023_fact__double,axiom,
    ! [N: 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))),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(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))),N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),N))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).

% fact_double
tff(fact_8024_fact__double,axiom,
    ! [N: 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))),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),N))),aa_nat_real(semiri2265585572941072030t_real,N)) ).

% fact_double
tff(fact_8025_binomial__code,axiom,
    ! [N: nat,K2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K2)),one_one_nat)),N),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K2)) ) ) ) ) ) ).

% binomial_code
tff(fact_8026_Maclaurin__lemma,axiom,
    ! [H: real,F2: fun_real_real,J2: fun_nat_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ? [B8: real] : aa_real_real(F2,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_vd(H),J2)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B8),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N)),aa_nat_real(semiri2265585572941072030t_real,N)))) ) ).

% Maclaurin_lemma
tff(fact_8027_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_na5723442673449079972x_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun5746040925223427547t_real(aTP_Lamm_ve(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_real(aa_nat480766339200631653x_real(Diff,zero_zero_nat),zero_zero_complex) ) ) ) ).

% Maclaurin_zero
tff(fact_8028_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_na6724076748692969250l_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_vf(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

% Maclaurin_zero
tff(fact_8029_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_nat_fun_rat_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun507281639146497029t_real(aTP_Lamm_vg(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_rat_real(aa_nat_fun_rat_real(Diff,zero_zero_nat),zero_zero_rat) ) ) ) ).

% Maclaurin_zero
tff(fact_8030_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_nat_fun_nat_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_vh(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).

% Maclaurin_zero
tff(fact_8031_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_nat_fun_int_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_vi(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).

% Maclaurin_zero
tff(fact_8032_sin__coeff__def,axiom,
    ! [X5: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X5))
       => ( aa_nat_real(sin_coeff,X5) = 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))),X5))
       => ( aa_nat_real(sin_coeff,X5) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X5),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X5)) ) ) ) ).

% sin_coeff_def
tff(fact_8033_fact__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M2)))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),N)) = 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,M2)),N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ) ) ).

% fact_diff_Suc
tff(fact_8034_sin__paired,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_vj,X)),aa_real_real(sin_real,X))) ).

% sin_paired
tff(fact_8035_and__int_Osimps,axiom,
    ! [K2: int,L: int] :
      ( ( ( 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(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),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))),K2))),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(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(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),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))),K2))),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(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.simps
tff(fact_8036_sin__zero,axiom,
    aa_complex_complex(sin_complex,zero_zero_complex) = zero_zero_complex ).

% sin_zero
tff(fact_8037_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_8038_bit_Oconj__zero__right,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_8039_bit_Oconj__zero__left,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,zero_zero_int),X) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_8040_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_8041_zero__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_and_eq
tff(fact_8042_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_8043_and__zero__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),zero_zero_nat) = zero_zero_nat ).

% and_zero_eq
tff(fact_8044_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_8045_and__numerals_I5_J,axiom,
    ! [X: num] : bit_se3949692690581998587nteger(numera6620942414471956472nteger(aa_num_num(bit0,X)),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% and_numerals(5)
tff(fact_8046_and__numerals_I5_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_8047_and__numerals_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_8048_and__numerals_I1_J,axiom,
    ! [Y: num] : bit_se3949692690581998587nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,Y))) = zero_z3403309356797280102nteger ).

% and_numerals(1)
tff(fact_8049_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = zero_zero_int ).

% and_numerals(1)
tff(fact_8050_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = zero_zero_nat ).

% and_numerals(1)
tff(fact_8051_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(3)
tff(fact_8052_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(3)
tff(fact_8053_and__int__rec,axiom,
    ! [K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),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))),K2))),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(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_8054_and__int__unfold,axiom,
    ! [K2: int,L: int] :
      ( ( ( ( K2 = zero_zero_int )
          | ( L = zero_zero_int ) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = zero_zero_int ) )
      & ( ~ ( ( K2 = zero_zero_int )
            | ( L = zero_zero_int ) )
       => ( ( ( K2 = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = L ) )
          & ( ( K2 != aa_int_int(uminus_uminus_int,one_one_int) )
           => ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
               => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = K2 ) )
              & ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
               => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),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(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% and_int_unfold
tff(fact_8055_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_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(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.elims
tff(fact_8056_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ? [T5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T5))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),X))
            & ( aa_real_real(sin_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vk,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_8057_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [T5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T5))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T5),X))
          & ( aa_real_real(sin_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vk,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_8058_sumr__cos__zero__one,axiom,
    ! [N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vl),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_8059_sin__cos__npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_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))),N)))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% sin_cos_npi
tff(fact_8060_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T5: real] : aa_real_real(sin_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vk,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ).

% Maclaurin_sin_expansion
tff(fact_8061_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_8062_semiring__norm_I15_J,axiom,
    ! [M2: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M2)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M2)),N)) ).

% semiring_norm(15)
tff(fact_8063_semiring__norm_I14_J,axiom,
    ! [M2: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,M2)),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M2),aa_num_num(bit1,N))) ).

% semiring_norm(14)
tff(fact_8064_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_8065_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K2)) = numera6690914467698888265omplex(aa_num_num(bit1,K2)) ).

% dbl_inc_simps(5)
tff(fact_8066_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K2)) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,K2)) ).

% dbl_inc_simps(5)
tff(fact_8067_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K2)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)) ).

% dbl_inc_simps(5)
tff(fact_8068_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_8069_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_8070_zdiv__numeral__Bit1,axiom,
    ! [V2: num,W2: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V2)),aa_num_int(numeral_numeral_int,W2)) ).

% zdiv_numeral_Bit1
tff(fact_8071_semiring__norm_I16_J,axiom,
    ! [M2: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M2)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N)),aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M2),N)))) ).

% semiring_norm(16)
tff(fact_8072_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_8073_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_8074_sin__npi2,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))) = zero_zero_real ).

% sin_npi2
tff(fact_8075_sin__npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_8076_dbl__inc__simps_I3_J,axiom,
    neg_nu5831290666863070958nteger(one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_8077_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_8078_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_8079_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_8080_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(4)
tff(fact_8081_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(4)
tff(fact_8082_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(6)
tff(fact_8083_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(6)
tff(fact_8084_Suc__0__and__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_8085_and__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_8086_div__Suc__eq__div__add3,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).

% div_Suc_eq_div_add3
tff(fact_8087_Suc__div__eq__add3__div__numeral,axiom,
    ! [M2: nat,V2: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M2)))),aa_num_nat(numeral_numeral_nat,V2)) = 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))),M2)),aa_num_nat(numeral_numeral_nat,V2)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_8088_mod__Suc__eq__mod__add3,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_8089_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M2: nat,V2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M2)))),aa_num_nat(numeral_numeral_nat,V2)) = 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))),M2)),aa_num_nat(numeral_numeral_nat,V2)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_8090_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_8091_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_8092_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_8093_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_8094_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_8095_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_8096_sin__2pi__minus,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(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(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_2pi_minus
tff(fact_8097_sin__periodic,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,X) ).

% sin_periodic
tff(fact_8098_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : bit_se3949692690581998587nteger(numera6620942414471956472nteger(aa_num_num(bit1,X)),numera6620942414471956472nteger(aa_num_num(bit1,Y))) = plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se3949692690581998587nteger(numera6620942414471956472nteger(X),numera6620942414471956472nteger(Y)))) ).

% and_numerals(7)
tff(fact_8099_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% and_numerals(7)
tff(fact_8100_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% and_numerals(7)
tff(fact_8101_zmod__numeral__Bit1,axiom,
    ! [V2: num,W2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V2))),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(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V2)),aa_num_int(numeral_numeral_int,W2)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_8102_sin__2npi,axiom,
    ! [N: 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,N))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_8103_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_8104_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_8105_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] : Y != aa_num_num(bit0,X23)
       => ~ ! [X32: num] : Y != aa_num_num(bit1,X32) ) ) ).

% num.exhaust
tff(fact_8106_numeral__Bit1,axiom,
    ! [N: num] : numera6620942414471956472nteger(aa_num_num(bit1,N)) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(N)),one_one_Code_integer) ).

% numeral_Bit1
tff(fact_8107_numeral__Bit1,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N))),one_one_rat) ).

% numeral_Bit1
tff(fact_8108_numeral__Bit1,axiom,
    ! [N: num] : numera1916890842035813515d_enat(aa_num_num(bit1,N)) = plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N),numera1916890842035813515d_enat(N)),one_on7984719198319812577d_enat) ).

% numeral_Bit1
tff(fact_8109_numeral__Bit1,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit1,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N))),one_one_complex) ).

% numeral_Bit1
tff(fact_8110_numeral__Bit1,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,N)) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N))),one_one_real) ).

% numeral_Bit1
tff(fact_8111_numeral__Bit1,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = 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,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).

% numeral_Bit1
tff(fact_8112_numeral__Bit1,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = 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,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ).

% numeral_Bit1
tff(fact_8113_eval__nat__numeral_I3_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))) ).

% eval_nat_numeral(3)
tff(fact_8114_cong__exp__iff__simps_I13_J,axiom,
    ! [M2: num,Q2: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_8115_cong__exp__iff__simps_I13_J,axiom,
    ! [M2: num,Q2: num,N: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_8116_cong__exp__iff__simps_I12_J,axiom,
    ! [M2: num,Q2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_8117_cong__exp__iff__simps_I12_J,axiom,
    ! [M2: num,Q2: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_8118_cong__exp__iff__simps_I10_J,axiom,
    ! [M2: num,Q2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_8119_cong__exp__iff__simps_I10_J,axiom,
    ! [M2: num,Q2: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_8120_numeral__code_I3_J,axiom,
    ! [N: num] : numera6620942414471956472nteger(aa_num_num(bit1,N)) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(N)),one_one_Code_integer) ).

% numeral_code(3)
tff(fact_8121_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,N)),aa_num_rat(numeral_numeral_rat,N))),one_one_rat) ).

% numeral_code(3)
tff(fact_8122_numeral__code_I3_J,axiom,
    ! [N: num] : numera1916890842035813515d_enat(aa_num_num(bit1,N)) = plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N),numera1916890842035813515d_enat(N)),one_on7984719198319812577d_enat) ).

% numeral_code(3)
tff(fact_8123_numeral__code_I3_J,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit1,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N))),one_one_complex) ).

% numeral_code(3)
tff(fact_8124_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,N)) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,N)),aa_num_real(numeral_numeral_real,N))),one_one_real) ).

% numeral_code(3)
tff(fact_8125_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = 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,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).

% numeral_code(3)
tff(fact_8126_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = 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,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ).

% numeral_code(3)
tff(fact_8127_power__numeral__odd,axiom,
    ! [Z2: complex,W2: num] : aa_nat_complex(power_power_complex(Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W2))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Z2),aa_nat_complex(power_power_complex(Z2),aa_num_nat(numeral_numeral_nat,W2)))),aa_nat_complex(power_power_complex(Z2),aa_num_nat(numeral_numeral_nat,W2))) ).

% power_numeral_odd
tff(fact_8128_power__numeral__odd,axiom,
    ! [Z2: real,W2: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),aa_num_nat(numeral_numeral_nat,W2)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),aa_num_nat(numeral_numeral_nat,W2))) ).

% power_numeral_odd
tff(fact_8129_power__numeral__odd,axiom,
    ! [Z2: rat,W2: num] : aa_nat_rat(power_power_rat(Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z2),aa_nat_rat(power_power_rat(Z2),aa_num_nat(numeral_numeral_nat,W2)))),aa_nat_rat(power_power_rat(Z2),aa_num_nat(numeral_numeral_nat,W2))) ).

% power_numeral_odd
tff(fact_8130_power__numeral__odd,axiom,
    ! [Z2: nat,W2: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z2),aa_num_nat(numeral_numeral_nat,W2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z2),aa_num_nat(numeral_numeral_nat,W2))) ).

% power_numeral_odd
tff(fact_8131_power__numeral__odd,axiom,
    ! [Z2: int,W2: num] : aa_nat_int(power_power_int(Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z2),aa_nat_int(power_power_int(Z2),aa_num_nat(numeral_numeral_nat,W2)))),aa_nat_int(power_power_int(Z2),aa_num_nat(numeral_numeral_nat,W2))) ).

% power_numeral_odd
tff(fact_8132_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_8133_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_8134_power3__eq__cube,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),A)),A) ).

% power3_eq_cube
tff(fact_8135_power3__eq__cube,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),A) ).

% power3_eq_cube
tff(fact_8136_power3__eq__cube,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),A) ).

% power3_eq_cube
tff(fact_8137_power3__eq__cube,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A)),A) ).

% power3_eq_cube
tff(fact_8138_power3__eq__cube,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),A) ).

% power3_eq_cube
tff(fact_8139_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_8140_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N) ).

% Suc3_eq_add_3
tff(fact_8141_num_Osize_I6_J,axiom,
    ! [X33: num] : aa_num_nat(size_size_num,aa_num_num(bit1,X33)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X33)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_8142_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_8143_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_8144_cong__exp__iff__simps_I11_J,axiom,
    ! [M2: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_8145_cong__exp__iff__simps_I11_J,axiom,
    ! [M2: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_8146_Suc__div__eq__add3__div,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M2)))),N) = 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))),M2)),N) ).

% Suc_div_eq_add3_div
tff(fact_8147_card__3__iff,axiom,
    ! [S3: set_complex] :
      ( ( finite_card_complex(S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X4: complex,Y5: complex,Z3: complex] :
          ( ( S3 = insert_complex(X4,insert_complex(Y5,insert_complex(Z3,bot_bot_set_complex))) )
          & ( X4 != Y5 )
          & ( Y5 != Z3 )
          & ( X4 != Z3 ) ) ) ).

% card_3_iff
tff(fact_8148_card__3__iff,axiom,
    ! [S3: set_list_nat] :
      ( ( finite_card_list_nat(S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X4: list_nat,Y5: list_nat,Z3: list_nat] :
          ( ( S3 = insert_list_nat(X4,insert_list_nat(Y5,insert_list_nat(Z3,bot_bot_set_list_nat))) )
          & ( X4 != Y5 )
          & ( Y5 != Z3 )
          & ( X4 != Z3 ) ) ) ).

% card_3_iff
tff(fact_8149_card__3__iff,axiom,
    ! [S3: set_set_nat] :
      ( ( finite_card_set_nat(S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X4: set_nat,Y5: set_nat,Z3: set_nat] :
          ( ( S3 = insert_set_nat(X4,insert_set_nat(Y5,insert_set_nat(Z3,bot_bot_set_set_nat))) )
          & ( X4 != Y5 )
          & ( Y5 != Z3 )
          & ( X4 != Z3 ) ) ) ).

% card_3_iff
tff(fact_8150_card__3__iff,axiom,
    ! [S3: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X4: nat,Y5: nat,Z3: nat] :
          ( ( S3 = insert_nat(X4,insert_nat(Y5,insert_nat(Z3,bot_bot_set_nat))) )
          & ( X4 != Y5 )
          & ( Y5 != Z3 )
          & ( X4 != Z3 ) ) ) ).

% card_3_iff
tff(fact_8151_card__3__iff,axiom,
    ! [S3: set_int] :
      ( ( finite_card_int(S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X4: int,Y5: int,Z3: int] :
          ( ( S3 = insert_int(X4,insert_int(Y5,insert_int(Z3,bot_bot_set_int))) )
          & ( X4 != Y5 )
          & ( Y5 != Z3 )
          & ( X4 != Z3 ) ) ) ).

% card_3_iff
tff(fact_8152_card__3__iff,axiom,
    ! [S3: set_real] :
      ( ( finite_card_real(S3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
    <=> ? [X4: real,Y5: real,Z3: real] :
          ( ( S3 = insert_real(X4,insert_real(Y5,insert_real(Z3,bot_bot_set_real))) )
          & ( X4 != Y5 )
          & ( Y5 != Z3 )
          & ( X4 != Z3 ) ) ) ).

% card_3_iff
tff(fact_8153_Suc__mod__eq__add3__mod,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M2)))),N) = 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))),M2)),N) ).

% Suc_mod_eq_add3_mod
tff(fact_8154_mod__exhaust__less__4,axiom,
    ! [M2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),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,M2),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,M2),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,M2),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_8155_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_8156_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != 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(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_8157_sin__coeff__Suc,axiom,
    ! [N: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,N)) = aa_real_real(divide_divide_real(aa_nat_real(cos_coeff,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))) ).

% sin_coeff_Suc
tff(fact_8158_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_8159_and__nat__unfold,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ( M2 = zero_zero_nat )
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M2),N) = zero_zero_nat ) )
      & ( ~ ( ( M2 = zero_zero_nat )
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M2),N) = 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,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),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,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% and_nat_unfold
tff(fact_8160_and__nat__rec,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M2),N) = 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))),M2))),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))),N))))),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,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_8161_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_8162_cos__coeff__Suc,axiom,
    ! [N: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,N)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,N))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))) ).

% cos_coeff_Suc
tff(fact_8163_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),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,N),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_8164_sin__pi__divide__n__gt__0,axiom,
    ! [N: 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))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_8165_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 )
       => ? [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_8166_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(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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_8167_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => ? [T5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T5))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),zero_zero_real))
            & ( aa_real_real(cos_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vm,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_8168_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: 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),N))
       => ? [T5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T5))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),X))
            & ( aa_real_real(cos_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vm,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_8169_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,X)))
      & ( aa_real_real(sin_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vk,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_8170_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_8171_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M2),aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M2))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M2),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit1,M2),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_8172_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M2),aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M2))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M2),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit1,M2),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_8173_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit1,M2),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,M2))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit1,M2),aa_num_num(bit1,N)) = unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit1,M2),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_8174_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit0,M2),aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M2))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit0,M2),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit0,M2),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_8175_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M2),aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M2))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M2),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit0,M2),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_8176_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit0,M2),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,M2))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit0,M2),aa_num_num(bit1,N)) = unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit0,M2),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_8177_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_8178_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_8179_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_8180_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_8181_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_8182_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_8183_abs__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_abs
tff(fact_8184_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_8185_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_8186_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_8187_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_8188_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_8189_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_8190_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_8191_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_8192_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_8193_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_8194_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_8195_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_8196_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_8197_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_8198_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_8199_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_8200_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_8201_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_8202_abs__numeral,axiom,
    ! [N: num] : abs_abs_Code_integer(numera6620942414471956472nteger(N)) = numera6620942414471956472nteger(N) ).

% abs_numeral
tff(fact_8203_abs__numeral,axiom,
    ! [N: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,N)) = aa_num_rat(numeral_numeral_rat,N) ).

% abs_numeral
tff(fact_8204_abs__numeral,axiom,
    ! [N: num] : aa_real_real(abs_abs_real,aa_num_real(numeral_numeral_real,N)) = aa_num_real(numeral_numeral_real,N) ).

% abs_numeral
tff(fact_8205_abs__numeral,axiom,
    ! [N: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,N) ).

% abs_numeral
tff(fact_8206_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_8207_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_8208_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) ).

% abs_mult_self_eq
tff(fact_8209_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_8210_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_8211_abs__add__abs,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) = aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_add_abs
tff(fact_8212_abs__add__abs,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_add_abs
tff(fact_8213_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_8214_abs__1,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_1
tff(fact_8215_abs__1,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_8216_abs__1,axiom,
    aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).

% abs_1
tff(fact_8217_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_8218_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_8219_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_8220_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_8221_abs__minus__cancel,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(uminus1351360451143612070nteger(A)) = abs_abs_Code_integer(A) ).

% abs_minus_cancel
tff(fact_8222_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_8223_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_8224_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_8225_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_8226_abs__minus,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(uminus1351360451143612070nteger(A)) = abs_abs_Code_integer(A) ).

% abs_minus
tff(fact_8227_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_8228_dvd__abs__iff,axiom,
    ! [M2: real,K2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M2),aa_real_real(abs_abs_real,K2)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M2),K2)) ) ).

% dvd_abs_iff
tff(fact_8229_dvd__abs__iff,axiom,
    ! [M2: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M2),aa_int_int(abs_abs_int,K2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M2),K2)) ) ).

% dvd_abs_iff
tff(fact_8230_dvd__abs__iff,axiom,
    ! [M2: code_integer,K2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M2),abs_abs_Code_integer(K2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M2),K2)) ) ).

% dvd_abs_iff
tff(fact_8231_dvd__abs__iff,axiom,
    ! [M2: rat,K2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M2),aa_rat_rat(abs_abs_rat,K2)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M2),K2)) ) ).

% dvd_abs_iff
tff(fact_8232_abs__dvd__iff,axiom,
    ! [M2: real,K2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(abs_abs_real,M2)),K2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M2),K2)) ) ).

% abs_dvd_iff
tff(fact_8233_abs__dvd__iff,axiom,
    ! [M2: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(abs_abs_int,M2)),K2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M2),K2)) ) ).

% abs_dvd_iff
tff(fact_8234_abs__dvd__iff,axiom,
    ! [M2: code_integer,K2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,abs_abs_Code_integer(M2)),K2))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M2),K2)) ) ).

% abs_dvd_iff
tff(fact_8235_abs__dvd__iff,axiom,
    ! [M2: rat,K2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(abs_abs_rat,M2)),K2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M2),K2)) ) ).

% abs_dvd_iff
tff(fact_8236_abs__of__nat,axiom,
    ! [N: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) ).

% abs_of_nat
tff(fact_8237_abs__of__nat,axiom,
    ! [N: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).

% abs_of_nat
tff(fact_8238_abs__of__nat,axiom,
    ! [N: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% abs_of_nat
tff(fact_8239_abs__of__nat,axiom,
    ! [N: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% abs_of_nat
tff(fact_8240_abs__bool__eq,axiom,
    ! [P2: bool] : aa_real_real(abs_abs_real,zero_n3304061248610475627l_real(P2)) = zero_n3304061248610475627l_real(P2) ).

% abs_bool_eq
tff(fact_8241_abs__bool__eq,axiom,
    ! [P2: bool] : aa_rat_rat(abs_abs_rat,zero_n2052037380579107095ol_rat(P2)) = zero_n2052037380579107095ol_rat(P2) ).

% abs_bool_eq
tff(fact_8242_abs__bool__eq,axiom,
    ! [P2: bool] : aa_int_int(abs_abs_int,aa_bool_int(zero_n2684676970156552555ol_int,P2)) = aa_bool_int(zero_n2684676970156552555ol_int,P2) ).

% abs_bool_eq
tff(fact_8243_abs__bool__eq,axiom,
    ! [P2: bool] : abs_abs_Code_integer(zero_n356916108424825756nteger(P2)) = zero_n356916108424825756nteger(P2) ).

% abs_bool_eq
tff(fact_8244_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_8245_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_8246_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_8247_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_8248_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_8249_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_8250_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_8251_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_8252_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_8253_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_8254_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_8255_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_8256_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_8257_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_8258_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_8259_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_8260_abs__neg__numeral,axiom,
    ! [N: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,N) ).

% abs_neg_numeral
tff(fact_8261_abs__neg__numeral,axiom,
    ! [N: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_num_real(numeral_numeral_real,N) ).

% abs_neg_numeral
tff(fact_8262_abs__neg__numeral,axiom,
    ! [N: num] : abs_abs_Code_integer(uminus1351360451143612070nteger(numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(N) ).

% abs_neg_numeral
tff(fact_8263_abs__neg__numeral,axiom,
    ! [N: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N))) = aa_num_rat(numeral_numeral_rat,N) ).

% abs_neg_numeral
tff(fact_8264_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_8265_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_8266_abs__neg__one,axiom,
    abs_abs_Code_integer(uminus1351360451143612070nteger(one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_8267_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_8268_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_8269_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_8270_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_8271_eq__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K2) = aa_nat_nat(suc,N) )
    <=> ( aa_num_nat(pred_numeral,K2) = N ) ) ).

% eq_numeral_Suc
tff(fact_8272_Suc__eq__numeral,axiom,
    ! [N: nat,K2: num] :
      ( ( aa_nat_nat(suc,N) = aa_num_nat(numeral_numeral_nat,K2) )
    <=> ( N = aa_num_nat(pred_numeral,K2) ) ) ).

% Suc_eq_numeral
tff(fact_8273_sum__abs,axiom,
    ! [F2: fun_int_int,A4: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F2,A4))),groups4538972089207619220nt_int(aTP_Lamm_vn(F2),A4))) ).

% sum_abs
tff(fact_8274_sum__abs,axiom,
    ! [F2: fun_nat_real,A4: 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(F2),A4))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vo(F2)),A4))) ).

% sum_abs
tff(fact_8275_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_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_8276_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(aa_rat_fun_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_8277_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(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_8278_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(aa_rat_fun_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_8279_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_8280_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) = uminus1351360451143612070nteger(A) ) ) ).

% abs_of_nonpos
tff(fact_8281_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_8282_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_8283_less__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(pred_numeral,K2))) ) ).

% less_Suc_numeral
tff(fact_8284_less__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(pred_numeral,K2)),N)) ) ).

% less_numeral_Suc
tff(fact_8285_pred__numeral__simps_I3_J,axiom,
    ! [K2: num] : aa_num_nat(pred_numeral,aa_num_num(bit1,K2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2)) ).

% pred_numeral_simps(3)
tff(fact_8286_le__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(pred_numeral,K2)),N)) ) ).

% le_numeral_Suc
tff(fact_8287_le__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_num_nat(pred_numeral,K2))) ) ).

% le_Suc_numeral
tff(fact_8288_diff__numeral__Suc,axiom,
    ! [K2: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K2)),N) ).

% diff_numeral_Suc
tff(fact_8289_diff__Suc__numeral,axiom,
    ! [N: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(pred_numeral,K2)) ).

% diff_Suc_numeral
tff(fact_8290_max__Suc__numeral,axiom,
    ! [N: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),aa_num_nat(pred_numeral,K2))) ).

% max_Suc_numeral
tff(fact_8291_max__numeral__Suc,axiom,
    ! [K2: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K2)),N)) ).

% max_numeral_Suc
tff(fact_8292_sum__abs__ge__zero,axiom,
    ! [F2: fun_int_int,A4: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_vn(F2),A4))) ).

% sum_abs_ge_zero
tff(fact_8293_sum__abs__ge__zero,axiom,
    ! [F2: fun_nat_real,A4: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vo(F2)),A4))) ).

% sum_abs_ge_zero
tff(fact_8294_minus__numeral__div__numeral,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M2,N))) ).

% minus_numeral_div_numeral
tff(fact_8295_numeral__div__minus__numeral,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M2,N))) ).

% numeral_div_minus_numeral
tff(fact_8296_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N)))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_8297_zero__less__power__abs__iff,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N)))
    <=> ( ( A != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_8298_zero__less__power__abs__iff,axiom,
    ! [A: rat,N: 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)),N)))
    <=> ( ( A != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_8299_zero__less__power__abs__iff,axiom,
    ! [A: int,N: 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)),N)))
    <=> ( ( A != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_8300_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_8301_sin__cos__squared__add3,axiom,
    ! [X: real] : aa_real_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_8302_dvd__numeral__simp,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_Pro4501881432599481050t_bool(unique6319869463603278526ux_int,unique5052692396658037445od_int(N,M2))) ) ).

% dvd_numeral_simp
tff(fact_8303_dvd__numeral__simp,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_Pro2232559814647311250t_bool(unique6322359934112328802ux_nat,unique5055182867167087721od_nat(N,M2))) ) ).

% dvd_numeral_simp
tff(fact_8304_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] : unique3479559517661332726nteger(M2,one) = produc1086072967326762835nteger(numera6620942414471956472nteger(M2),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_8305_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] : unique5052692396658037445od_int(M2,one) = product_Pair_int_int(aa_num_int(numeral_numeral_int,M2),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_8306_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] : unique5055182867167087721od_nat(M2,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,M2)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_8307_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(3)
tff(fact_8308_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_8309_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_8310_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(4)
tff(fact_8311_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_8312_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_8313_one__div__minus__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,N))) ).

% one_div_minus_numeral
tff(fact_8314_minus__one__div__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,N))) ).

% minus_one_div_numeral
tff(fact_8315_cos__two__pi,axiom,
    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))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_8316_cos__periodic,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,X) ).

% cos_periodic
tff(fact_8317_cos__2pi__minus,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(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,X) ).

% cos_2pi_minus
tff(fact_8318_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_bit0
tff(fact_8319_cos__npi,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% cos_npi
tff(fact_8320_cos__npi2,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% cos_npi2
tff(fact_8321_cos__2npi,axiom,
    ! [N: nat] : aa_real_real(cos_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,N))),pi)) = one_one_real ).

% cos_2npi
tff(fact_8322_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_8323_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_8324_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_8325_cos__pi__eq__zero,axiom,
    ! [M2: nat] : aa_real_real(cos_real,aa_real_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))),M2))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_8326_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_8327_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_8328_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_8329_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_8330_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_8331_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_8332_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_8333_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_8334_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_8335_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(abs_abs_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
tff(fact_8336_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_8337_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_8338_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_8339_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_8340_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_8341_abs__mult,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_mult
tff(fact_8342_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_8343_abs__one,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_one
tff(fact_8344_abs__one,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_8345_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_8346_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_8347_abs__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B)) = aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A)) ).

% abs_minus_commute
tff(fact_8348_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_8349_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)) ).

% abs_minus_commute
tff(fact_8350_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_8351_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_8352_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_8353_abs__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( abs_abs_Code_integer(X) = abs_abs_Code_integer(Y) )
    <=> ( ( X = Y )
        | ( X = uminus1351360451143612070nteger(Y) ) ) ) ).

% abs_eq_iff
tff(fact_8354_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_8355_dvd__if__abs__eq,axiom,
    ! [L: real,K2: real] :
      ( ( aa_real_real(abs_abs_real,L) = aa_real_real(abs_abs_real,K2) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),K2)) ) ).

% dvd_if_abs_eq
tff(fact_8356_dvd__if__abs__eq,axiom,
    ! [L: int,K2: int] :
      ( ( aa_int_int(abs_abs_int,L) = aa_int_int(abs_abs_int,K2) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K2)) ) ).

% dvd_if_abs_eq
tff(fact_8357_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K2: code_integer] :
      ( ( abs_abs_Code_integer(L) = abs_abs_Code_integer(K2) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,L),K2)) ) ).

% dvd_if_abs_eq
tff(fact_8358_dvd__if__abs__eq,axiom,
    ! [L: rat,K2: rat] :
      ( ( aa_rat_rat(abs_abs_rat,L) = aa_rat_rat(abs_abs_rat,K2) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,L),K2)) ) ).

% dvd_if_abs_eq
tff(fact_8359_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_8360_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_8361_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_8362_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_8363_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_8364_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_8365_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_8366_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_8367_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_8368_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_8369_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_8370_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_8371_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_8372_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(plus_plus_real(A),B))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))) ).

% abs_triangle_ineq
tff(fact_8373_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(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))) ).

% abs_triangle_ineq
tff(fact_8374_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_8375_abs__mult__less,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),C2))
     => ( 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,C2),D))) ) ) ).

% abs_mult_less
tff(fact_8376_abs__mult__less,axiom,
    ! [A: real,C2: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),C2))
     => ( 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,C2),D))) ) ) ).

% abs_mult_less
tff(fact_8377_abs__mult__less,axiom,
    ! [A: rat,C2: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C2))
     => ( 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(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),D))) ) ) ).

% abs_mult_less
tff(fact_8378_abs__mult__less,axiom,
    ! [A: int,C2: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),C2))
     => ( 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,C2),D))) ) ) ).

% abs_mult_less
tff(fact_8379_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_8380_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_8381_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(aa_rat_fun_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(aa_rat_fun_rat_rat(minus_minus_rat,B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_8382_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_8383_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(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B)))) ).

% abs_triangle_ineq3
tff(fact_8384_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_8385_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(aa_rat_fun_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(aa_rat_fun_rat_rat(minus_minus_rat,A),B)))) ).

% abs_triangle_ineq3
tff(fact_8386_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_8387_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B)))) ).

% abs_triangle_ineq2
tff(fact_8388_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_8389_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(minus_minus_rat,A),B)))) ).

% abs_triangle_ineq2
tff(fact_8390_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_8391_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% nonzero_abs_divide
tff(fact_8392_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% nonzero_abs_divide
tff(fact_8393_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_8394_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,uminus1351360451143612070nteger(A)),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)) ) ) ).

% abs_leI
tff(fact_8395_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_8396_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_8397_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_8398_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,uminus1351360451143612070nteger(A)),B)) ) ).

% abs_le_D2
tff(fact_8399_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_8400_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_8401_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_8402_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,uminus1351360451143612070nteger(A)),B)) ) ) ).

% abs_le_iff
tff(fact_8403_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_8404_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_8405_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_8406_abs__ge__minus__self,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),abs_abs_Code_integer(A))) ).

% abs_ge_minus_self
tff(fact_8407_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_8408_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_8409_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_8410_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_8411_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,uminus1351360451143612070nteger(A)),B)) ) ) ).

% abs_less_iff
tff(fact_8412_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_8413_polar__Ex,axiom,
    ! [X: real,Y: real] :
    ? [R3: real,A3: real] :
      ( ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_real_real(cos_real,A3)) )
      & ( Y = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_real_real(sin_real,A3)) ) ) ).

% polar_Ex
tff(fact_8414_numeral__eq__Suc,axiom,
    ! [K2: num] : aa_num_nat(numeral_numeral_nat,K2) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K2)) ).

% numeral_eq_Suc
tff(fact_8415_sin__cos__le1,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))))),one_one_real)) ).

% sin_cos_le1
tff(fact_8416_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),E2)) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_8417_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),E2)) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_8418_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_8419_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_8420_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(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,Y)),X) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_8421_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_8422_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_8423_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_8424_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(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_8425_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_8426_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_8427_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_8428_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_8429_abs__minus__le__zero,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(abs_abs_Code_integer(A))),zero_z3403309356797280102nteger)) ).

% abs_minus_le_zero
tff(fact_8430_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_8431_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_8432_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_8433_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 = uminus1351360451143612070nteger(A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_8434_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_8435_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_8436_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_8437_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 = uminus1351360451143612070nteger(B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_8438_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_8439_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_8440_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(divide_divide_real(aa_real_real(abs_abs_real,X)),Y) = aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(X),Y)) ) ) ).

% abs_div_pos
tff(fact_8441_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(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(abs_abs_rat,X)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% abs_div_pos
tff(fact_8442_zero__le__power__abs,axiom,
    ! [A: code_integer,N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N))) ).

% zero_le_power_abs
tff(fact_8443_zero__le__power__abs,axiom,
    ! [A: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N))) ).

% zero_le_power_abs
tff(fact_8444_zero__le__power__abs,axiom,
    ! [A: rat,N: 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)),N))) ).

% zero_le_power_abs
tff(fact_8445_zero__le__power__abs,axiom,
    ! [A: int,N: 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)),N))) ).

% zero_le_power_abs
tff(fact_8446_abs__if__raw,axiom,
    ! [X5: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X5) = aa_int_int(uminus_uminus_int,X5) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X5) = X5 ) ) ) ).

% abs_if_raw
tff(fact_8447_abs__if__raw,axiom,
    ! [X5: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X5) = aa_real_real(uminus_uminus_real,X5) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X5) = X5 ) ) ) ).

% abs_if_raw
tff(fact_8448_abs__if__raw,axiom,
    ! [X5: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X5),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X5) = uminus1351360451143612070nteger(X5) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X5),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X5) = X5 ) ) ) ).

% abs_if_raw
tff(fact_8449_abs__if__raw,axiom,
    ! [X5: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,X5) = aa_rat_rat(uminus_uminus_rat,X5) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,X5) = X5 ) ) ) ).

% abs_if_raw
tff(fact_8450_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_8451_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_8452_abs__if,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(A) = 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_8453_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_8454_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_8455_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_8456_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) = uminus1351360451143612070nteger(A) ) ) ).

% abs_of_neg
tff(fact_8457_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_8458_sin__add,axiom,
    ! [X: real,Y: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(X),Y)) = aa_real_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_8459_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R)),X))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),plus_p5714425477246183910nteger(A,R))) ) ) ).

% abs_diff_le_iff
tff(fact_8460_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),A))),R))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(A),R)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(plus_plus_real(A),R))) ) ) ).

% abs_diff_le_iff
tff(fact_8461_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),A))),R))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),R)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R))) ) ) ).

% abs_diff_le_iff
tff(fact_8462_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R: 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))),R))
    <=> ( 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),R)),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),R))) ) ) ).

% abs_diff_le_iff
tff(fact_8463_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C2: 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(C2,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,C2)),abs_abs_Code_integer(minus_8373710615458151222nteger(B,D))))) ).

% abs_diff_triangle_ineq
tff(fact_8464_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C2: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(C2),D)))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),C2))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_8465_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C2: 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(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),D)))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C2))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_8466_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C2: 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,C2),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),C2))),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_8467_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_8468_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(minus_minus_real(A),B))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))) ).

% abs_triangle_ineq4
tff(fact_8469_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(aa_rat_fun_rat_rat(minus_minus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))) ).

% abs_triangle_ineq4
tff(fact_8470_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_8471_sin__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(X),Y)) = aa_real_real(minus_minus_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_diff
tff(fact_8472_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R)),X))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),plus_p5714425477246183910nteger(A,R))) ) ) ).

% abs_diff_less_iff
tff(fact_8473_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),A))),R))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),R)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(plus_plus_real(A),R))) ) ) ).

% abs_diff_less_iff
tff(fact_8474_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),A))),R))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),R)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R))) ) ) ).

% abs_diff_less_iff
tff(fact_8475_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R: 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))),R))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R)),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),R))) ) ) ).

% abs_diff_less_iff
tff(fact_8476_pred__numeral__def,axiom,
    ! [K2: num] : aa_num_nat(pred_numeral,K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K2)),one_one_nat) ).

% pred_numeral_def
tff(fact_8477_lessThan__nat__numeral,axiom,
    ! [K2: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K2)) = insert_nat(aa_num_nat(pred_numeral,K2),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K2))) ).

% lessThan_nat_numeral
tff(fact_8478_atMost__nat__numeral,axiom,
    ! [K2: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K2)) = insert_nat(aa_num_nat(numeral_numeral_nat,K2),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K2))) ).

% atMost_nat_numeral
tff(fact_8479_cos__add,axiom,
    ! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(X),Y)) = aa_real_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_8480_cos__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(X),Y)) = aa_real_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_8481_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_8482_abs__add__one__gt__zero,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(one_one_real),aa_real_real(abs_abs_real,X)))) ).

% abs_add_one_gt_zero
tff(fact_8483_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(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(abs_abs_rat,X)))) ).

% abs_add_one_gt_zero
tff(fact_8484_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_8485_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_8486_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_8487_fact__numeral,axiom,
    ! [K2: num] : aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_8488_fact__numeral,axiom,
    ! [K2: num] : semiri4449623510593786356d_enat(aa_num_nat(numeral_numeral_nat,K2)) = times_7803423173614009249d_enat(numera1916890842035813515d_enat(K2),semiri4449623510593786356d_enat(aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_8489_fact__numeral,axiom,
    ! [K2: num] : aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,K2)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(K2)),aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_8490_fact__numeral,axiom,
    ! [K2: num] : aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,K2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,K2)),aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_8491_fact__numeral,axiom,
    ! [K2: num] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_8492_fact__numeral,axiom,
    ! [K2: num] : aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,K2)),aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_8493_sin__double,axiom,
    ! [X: complex] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,X))),aa_complex_complex(cos_complex,X)) ).

% sin_double
tff(fact_8494_sin__double,axiom,
    ! [X: 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(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,X))),aa_real_real(cos_real,X)) ).

% sin_double
tff(fact_8495_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_8496_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(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_8497_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_8498_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_8499_divmod__int__def,axiom,
    ! [M2: num,N: num] : unique5052692396658037445od_int(M2,N) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N))) ).

% divmod_int_def
tff(fact_8500_abs__sqrt__wlog,axiom,
    ! [P2: fun_Co8310303586393106575r_bool,X: code_integer] :
      ( ! [X3: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X3))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P2,X3),aa_nat_Code_integer(power_8256067586552552935nteger(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P2,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_8501_abs__sqrt__wlog,axiom,
    ! [P2: fun_re1255401033344146673l_bool,X: real] :
      ( ! [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P2,X3),aa_nat_real(aa_real_fun_nat_real(power_power_real,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P2,aa_real_real(abs_abs_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_8502_abs__sqrt__wlog,axiom,
    ! [P2: fun_rat_fun_rat_bool,X: rat] :
      ( ! [X3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,X3),aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(P2,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_8503_abs__sqrt__wlog,axiom,
    ! [P2: fun_int_fun_int_bool,X: int] :
      ( ! [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
         => pp(aa_int_bool(aa_int_fun_int_bool(P2,X3),aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(P2,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_8504_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_8505_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(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> 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_8506_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_8507_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_8508_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_8509_abs__square__le__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_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_8510_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_8511_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_8512_divmod__def,axiom,
    ! [M2: num,N: num] : unique3479559517661332726nteger(M2,N) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M2),numera6620942414471956472nteger(N)),modulo364778990260209775nteger(numera6620942414471956472nteger(M2),numera6620942414471956472nteger(N))) ).

% divmod_def
tff(fact_8513_divmod__def,axiom,
    ! [M2: num,N: num] : unique5052692396658037445od_int(M2,N) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M2)),aa_num_int(numeral_numeral_int,N))) ).

% divmod_def
tff(fact_8514_divmod__def,axiom,
    ! [M2: num,N: num] : unique5055182867167087721od_nat(M2,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N))) ).

% divmod_def
tff(fact_8515_divmod_H__nat__def,axiom,
    ! [M2: num,N: num] : unique5055182867167087721od_nat(M2,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N))) ).

% divmod'_nat_def
tff(fact_8516_power__mono__even,axiom,
    ! [N: 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))),N))
     => ( 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),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))) ) ) ).

% power_mono_even
tff(fact_8517_power__mono__even,axiom,
    ! [N: 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))),N))
     => ( 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(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ).

% power_mono_even
tff(fact_8518_power__mono__even,axiom,
    ! [N: 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))),N))
     => ( 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),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).

% power_mono_even
tff(fact_8519_power__mono__even,axiom,
    ! [N: 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))),N))
     => ( 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),N)),aa_nat_int(power_power_int(B),N))) ) ) ).

% power_mono_even
tff(fact_8520_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: fun_co8153928105989009288nteger,A: fun_co8153928105989009288nteger,B: code_integer,Delta: code_integer] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(X,I3))) )
     => ( ( groups6621422865394947399nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_com4217951221296830281nteger(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups6621422865394947399nteger(aa_fun5267417455630836558nteger(aTP_Lamm_vp(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8521_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_re6978948838062957830nteger,A: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(X,I3))) )
     => ( ( groups7713935264441627589nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_real_Code_integer(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_vq(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8522_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(X,I3))) )
     => ( ( groups7501900531339628137nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: nat] :
              ( pp(aa_set_nat_bool(member_nat(I3),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_nat_Code_integer(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_vr(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8523_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(X,I3))) )
     => ( ( groups7873554091576472773nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_int_Code_integer(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_vs(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8524_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: fun_complex_real,A: fun_complex_real,B: real,Delta: real] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(X,I3))) )
     => ( ( groups5808333547571424918x_real(X,I5) = one_one_real )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_complex_real(A,I3)),B))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_vt(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8525_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_real_real,A: fun_real_real,B: real,Delta: real] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(X,I3))) )
     => ( ( groups8097168146408367636l_real(X,I5) = one_one_real )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(A,I3)),B))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_vu(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8526_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_real,A: fun_int_real,B: real,Delta: real] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(X,I3))) )
     => ( ( groups8778361861064173332t_real(X,I5) = one_one_real )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_int_real(A,I3)),B))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_vv(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8527_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: fun_complex_rat,A: fun_complex_rat,B: rat,Delta: rat] :
      ( ! [I3: complex] :
          ( pp(aa_set_complex_bool(member_complex(I3),I5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(X,I3))) )
     => ( ( groups5058264527183730370ex_rat(X,I5) = one_one_rat )
       => ( ! [I3: complex] :
              ( pp(aa_set_complex_bool(member_complex(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_complex_rat(A,I3)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_vw(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8528_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_real_rat,A: fun_real_rat,B: rat,Delta: rat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),I5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(X,I3))) )
     => ( ( groups1300246762558778688al_rat(X,I5) = one_one_rat )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_real_rat(A,I3)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_vx(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8529_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),I5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(X,I3))) )
     => ( ( groups2906978787729119204at_rat(X,I5) = one_one_rat )
       => ( ! [I3: nat] :
              ( pp(aa_set_nat_bool(member_nat(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(A,I3)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_vy(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8530_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_8531_cos__plus__cos,axiom,
    ! [W2: real,Z2: real] : aa_real_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(divide_divide_real(aa_real_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(divide_divide_real(aa_real_real(minus_minus_real(W2),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_8532_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_8533_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(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(cos_real,aa_real_real(minus_minus_real(W2),Z2))),aa_real_real(cos_real,aa_real_real(plus_plus_real(W2),Z2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_8534_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_8535_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_8536_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(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,W2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_real) ).

% cos_double_cos
tff(fact_8537_cos__treble__cos,axiom,
    ! [X: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),X)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),aa_complex_complex(cos_complex,X))) ).

% cos_treble_cos
tff(fact_8538_cos__treble__cos,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),X)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_real_real(cos_real,X))) ).

% cos_treble_cos
tff(fact_8539_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_8540_cos__diff__cos,axiom,
    ! [W2: real,Z2: real] : aa_real_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(divide_divide_real(aa_real_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(divide_divide_real(aa_real_real(minus_minus_real(Z2),W2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_8541_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_8542_sin__diff__sin,axiom,
    ! [W2: real,Z2: real] : aa_real_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(divide_divide_real(aa_real_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(divide_divide_real(aa_real_real(plus_plus_real(W2),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_8543_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_8544_sin__plus__sin,axiom,
    ! [W2: real,Z2: real] : aa_real_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(divide_divide_real(aa_real_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(divide_divide_real(aa_real_real(minus_minus_real(W2),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_8545_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_8546_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(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(W2),Z2))),aa_real_real(sin_real,aa_real_real(minus_minus_real(W2),Z2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_8547_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_8548_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(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(W2),Z2))),aa_real_real(sin_real,aa_real_real(minus_minus_real(W2),Z2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_8549_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_8550_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(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(cos_real,aa_real_real(minus_minus_real(W2),Z2))),aa_real_real(cos_real,aa_real_real(plus_plus_real(W2),Z2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_8551_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,X)))
      & ( aa_real_real(cos_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vm,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_8552_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ( ? [X4: nat] : X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X4)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)
        | ? [X4: nat] : X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X4)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_8553_sin__expansion__lemma,axiom,
    ! [X: real,M2: nat] : aa_real_real(sin_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M2))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_8554_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 )
       => ? [N3: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_8555_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(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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_8556_cos__expansion__lemma,axiom,
    ! [X: real,M2: nat] : aa_real_real(cos_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M2))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_8557_cos__paired,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_vz,X)),aa_real_real(cos_real,X))) ).

% cos_paired
tff(fact_8558_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T5: real] :
          ( 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),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,T5) )
          & ( Y = aa_real_real(sin_real,T5) ) ) ) ).

% sincos_total_2pi_le
tff(fact_8559_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T5))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),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,T5) )
               => ( Y != aa_real_real(sin_real,T5) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_8560_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(aa_real_fun_nat_real(aTP_Lamm_wa,X)) ) ).

% monoseq_arctan_series
tff(fact_8561_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(aa_real_fun_nat_real(aTP_Lamm_wb,X)) ) ).

% summable_arctan_series
tff(fact_8562_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(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(minus_minus_real(aa_real_real(ln_ln_real,aa_real_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(aa_real_fun_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_8563_pi__series,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_wc) ).

% pi_series
tff(fact_8564_and__int_Opsimps,axiom,
    ! [K2: int,L: int] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(K2,L)))
     => ( ( ( 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(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),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))),K2))),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(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(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),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))),K2))),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(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_8565_tan__npi,axiom,
    ! [N: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_8566_tan__periodic__n,axiom,
    ! [X: real,N: num] : aa_real_real(tan_real,aa_real_real(plus_plus_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,N)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_n
tff(fact_8567_tan__periodic__nat,axiom,
    ! [X: real,N: nat] : aa_real_real(tan_real,aa_real_real(plus_plus_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_nat
tff(fact_8568_tan__periodic,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(plus_plus_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic
tff(fact_8569_abs__zmult__eq__1,axiom,
    ! [M2: int,N: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,M2),N)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M2) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_8570_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(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,X)),aa_int_int(abs_abs_int,Y)) ) ) ).

% abs_div
tff(fact_8571_summable__rabs__comparison__test,axiom,
    ! [F2: fun_nat_real,G2: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N6),N3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F2,N3))),aa_nat_real(G2,N3))) )
     => ( summable_real(G2)
       => summable_real(aTP_Lamm_vo(F2)) ) ) ).

% summable_rabs_comparison_test
tff(fact_8572_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(plus_plus_real(aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_8573_zdvd__mult__cancel1,axiom,
    ! [M2: int,N: int] :
      ( ( M2 != 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,M2),N)),M2))
      <=> ( aa_int_int(abs_abs_int,N) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_8574_sum__pos__lt__pair,axiom,
    ! [F2: fun_nat_real,K2: nat] :
      ( summable_real(F2)
     => ( ! [D5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(F2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D5)))),aa_nat_real(F2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D5)),one_one_nat))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F2),aa_nat_set_nat(set_ord_lessThan_nat,K2))),suminf_real(F2))) ) ) ).

% sum_pos_lt_pair
tff(fact_8575_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(ln_ln_real,X)) ) ) ).

% ln_realpow
tff(fact_8576_summable__power__series,axiom,
    ! [F2: fun_nat_real,Z2: real] :
      ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F2,I3)),one_one_real))
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F2,I3)))
       => ( 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_qj(F2),Z2)) ) ) ) ) ).

% summable_power_series
tff(fact_8577_nat__intermed__int__val,axiom,
    ! [M2: nat,N: nat,F2: fun_nat_int,K2: int] :
      ( ! [I3: nat] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),I3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N)) )
         => 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(F2,aa_nat_nat(suc,I3))),aa_nat_int(F2,I3)))),one_one_int)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F2,M2)),K2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),aa_nat_int(F2,N)))
           => ? [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),I3))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
                & ( aa_nat_int(F2,I3) = K2 ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_8578_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_8579_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_8580_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(aa_real_fun_nat_real(aTP_Lamm_wd,X)) ) ) ) ).

% ln_series
tff(fact_8581_nat__ivt__aux,axiom,
    ! [N: nat,F2: fun_nat_int,K2: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => 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(F2,aa_nat_nat(suc,I3))),aa_nat_int(F2,I3)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F2,zero_zero_nat)),K2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),aa_nat_int(F2,N)))
         => ? [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
              & ( aa_nat_int(F2,I3) = K2 ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_8582_nat0__intermed__int__val,axiom,
    ! [N: nat,F2: fun_nat_int,K2: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => 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(F2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),one_one_nat))),aa_nat_int(F2,I3)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F2,zero_zero_nat)),K2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K2),aa_nat_int(F2,N)))
         => ? [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
              & ( aa_nat_int(F2,I3) = K2 ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_8583_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(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(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(aa_real_fun_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(minus_minus_real(one_one_real),X)))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_8584_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_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(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_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(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_8585_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa2) = Y )
     => ( pp(aa_Pro4501881432599481050t_bool(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(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) )
           => ~ pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(X,Xa2))) ) ) ) ).

% and_int.pelims
tff(fact_8586_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(aa_real_fun_nat_real(aTP_Lamm_wb,X)) ) ) ).

% arctan_series
tff(fact_8587_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T5)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(exp_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_we,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(exp_real,T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_8588_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(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(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_8589_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(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(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_8590_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: 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,N))),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(one_one_real),aa_real_real(divide_divide_real(X),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,X))) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_8591_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(one_one_real),aa_real_real(divide_divide_real(X),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_8592_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(plus_plus_real(aa_real_real(arctan,X)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),Y)),aa_real_real(minus_minus_real(one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)))) ) ) ) ).

% arctan_add
tff(fact_8593_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,X)))
      & ( aa_real_real(exp_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_we,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(exp_real,T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).

% Maclaurin_exp_le
tff(fact_8594_machin__Euler,axiom,
    aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_8595_machin,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))))))) ).

% machin
tff(fact_8596_tanh__real__altdef,axiom,
    ! [X: real] : aa_real_real(tanh_real,X) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)))),aa_real_real(plus_plus_real(one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)))) ).

% tanh_real_altdef
tff(fact_8597_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(sin_real,X)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vk,X)),aa_nat_set_nat(set_ord_lessThan_nat,N))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,X)),N)))) ).

% Maclaurin_sin_bound
tff(fact_8598_divmod__BitM__2__eq,axiom,
    ! [M2: num] : unique5052692396658037445od_int(aa_num_num(bitM,M2),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,M2)),one_one_int),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_8599_pred__numeral__simps_I2_J,axiom,
    ! [K2: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,K2)) ).

% pred_numeral_simps(2)
tff(fact_8600_sin__npi__int,axiom,
    ! [N: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = zero_zero_real ).

% sin_npi_int
tff(fact_8601_tan__periodic__int,axiom,
    ! [X: real,I2: int] : aa_real_real(tan_real,aa_real_real(plus_plus_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I2)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_int
tff(fact_8602_sin__int__2pin,axiom,
    ! [N: 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,N))) = zero_zero_real ).

% sin_int_2pin
tff(fact_8603_cos__int__2pin,axiom,
    ! [N: int] : aa_real_real(cos_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,N))) = one_one_real ).

% cos_int_2pin
tff(fact_8604_cos__npi__int,axiom,
    ! [N: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = one_one_real ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% cos_npi_int
tff(fact_8605_real__scaleR__def,axiom,
    ! [A: real,X: real] : aa_real_real(real_V1485227260804924795R_real(A),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),X) ).

% real_scaleR_def
tff(fact_8606_divide__real__def,axiom,
    ! [X: real,Y: real] : aa_real_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_8607_eval__nat__numeral_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,N))) ).

% eval_nat_numeral(2)
tff(fact_8608_BitM__plus__one,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bitM,N)),one) = aa_num_num(bit0,N) ).

% BitM_plus_one
tff(fact_8609_one__plus__BitM,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bitM,N)) = aa_num_num(bit0,N) ).

% one_plus_BitM
tff(fact_8610_forall__pos__mono__1,axiom,
    ! [P2: fun_real_bool,E: real] :
      ( ! [D5: real,E2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D5),E2))
         => ( pp(aa_real_bool(P2,D5))
           => pp(aa_real_bool(P2,E2)) ) )
     => ( ! [N3: nat] : pp(aa_real_bool(P2,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N3)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(P2,E)) ) ) ) ).

% forall_pos_mono_1
tff(fact_8611_forall__pos__mono,axiom,
    ! [P2: fun_real_bool,E: real] :
      ( ! [D5: real,E2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D5),E2))
         => ( pp(aa_real_bool(P2,D5))
           => pp(aa_real_bool(P2,E2)) ) )
     => ( ! [N3: nat] :
            ( ( N3 != zero_zero_nat )
           => pp(aa_real_bool(P2,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N3)))) )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(P2,E)) ) ) ) ).

% forall_pos_mono
tff(fact_8612_real__arch__inverse,axiom,
    ! [E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
    <=> ? [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))),E)) ) ) ).

% real_arch_inverse
tff(fact_8613_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I: int] : X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),pi) ) ).

% sin_zero_iff_int2
tff(fact_8614_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ? [X4: int] : X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X4)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_8615_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ? [I: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I))
          & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_8616_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I))
          & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_8617_complex__unimodular__polar,axiom,
    ! [Z2: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z2) = one_one_real )
     => ~ ! [T5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T5))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),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,T5)),aa_real_real(sin_real,T5)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_8618_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K: int] : aa_real_real(arccos,aa_real_real(cos_real,Theta)) != aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Theta),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,K)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_8619_log__base__10__eq1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(log2(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(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_8620_complex__scaleR,axiom,
    ! [R: real,A: real,B: real] : aa_complex_complex(real_V2046097035970521341omplex(R),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),B)) ).

% complex_scaleR
tff(fact_8621_complex__mult,axiom,
    ! [A: real,B: real,C2: real,D: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C2),D)) = aa_real_complex(complex2(aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_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),C2))) ).

% complex_mult
tff(fact_8622_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(log2(A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(plus_plus_real(aa_real_real(log2(A),X)),aa_real_real(log2(A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_8623_log__nat__power,axiom,
    ! [X: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(log2(B),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log2(B),X)) ) ) ).

% log_nat_power
tff(fact_8624_log__of__power__less,axiom,
    ! [M2: nat,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
     => ( 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),M2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log2(B),aa_nat_real(semiri5074537144036343181t_real,M2))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_less
tff(fact_8625_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(log2(A),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B)),aa_real_real(ln_ln_real,A))),aa_real_real(log2(B),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_8626_log__of__power__le,axiom,
    ! [M2: nat,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
     => ( 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),M2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log2(B),aa_nat_real(semiri5074537144036343181t_real,M2))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_le
tff(fact_8627_le__log2__of__power,axiom,
    ! [N: nat,M2: 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))),N)),M2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log2(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M2)))) ) ).

% le_log2_of_power
tff(fact_8628_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_wf(L)),Qr) ).

% divmod_step_nat_def
tff(fact_8629_log2__of__power__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log2(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M2))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_less
tff(fact_8630_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_wg(L),Qr) ).

% divmod_step_int_def
tff(fact_8631_log2__of__power__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log2(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M2))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_le
tff(fact_8632_log__base__10__eq2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(log2(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(log2(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_8633_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K2: nat,N: 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),K2))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log2(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),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),N)),K2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_8634_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K2: 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)),K2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
       => ( 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(log2(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_8635_ceiling__log2__div2,axiom,
    ! [N: 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))),N))
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log2(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(log2(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,N),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_8636_divmod__nat__if,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( ( N = zero_zero_nat )
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) )
       => ( divmod_nat(M2,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M2) ) )
      & ( ~ ( ( N = zero_zero_nat )
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) )
       => ( divmod_nat(M2,N) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_wh),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_8637_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K2: nat,N: 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),K2))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log2(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_nat_int(semiri1314217659103216013at_int,N) )
        <=> ( 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)),K2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_8638_divide__complex__def,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,Y)) ).

% divide_complex_def
tff(fact_8639_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_wi) ).

% prod_encode_def
tff(fact_8640_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : adjust_div(Qr) = produc8211389475949308722nt_int(aTP_Lamm_wj,Qr) ).

% Divides.adjust_div_def
tff(fact_8641_divmod__nat__def,axiom,
    ! [M2: nat,N: nat] : divmod_nat(M2,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)) ).

% divmod_nat_def
tff(fact_8642_floor__log2__div2,axiom,
    ! [N: 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))),N))
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log2(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(log2(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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_8643_floor__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K2: 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)),K2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
       => ( 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(log2(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_8644_cot__periodic,axiom,
    ! [X: real] : aa_real_real(cot_real,aa_real_real(plus_plus_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,X) ).

% cot_periodic
tff(fact_8645_modulo__int__unfold,axiom,
    ! [L: int,K2: int,N: nat,M2: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K2) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,M2)) ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K2) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = 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,M2),N))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K2) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = 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,N),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M2)))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_8646_dvd__mult__sgn__iff,axiom,
    ! [L: int,K2: int,R: 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,K2),aa_int_int(sgn_sgn_int,R))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K2))
        | ( R = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_8647_dvd__sgn__mult__iff,axiom,
    ! [L: int,R: int,K2: 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,R)),K2)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K2))
        | ( R = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_8648_mult__sgn__dvd__iff,axiom,
    ! [L: int,R: int,K2: 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,R))),K2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K2))
        & ( ( R = zero_zero_int )
         => ( K2 = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_8649_sgn__mult__dvd__iff,axiom,
    ! [R: int,L: int,K2: 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,R)),L)),K2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K2))
        & ( ( R = zero_zero_int )
         => ( K2 = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_8650_cot__npi,axiom,
    ! [N: nat] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_8651_int__sgnE,axiom,
    ! [K2: int] :
      ~ ! [N3: nat,L3: int] : K2 != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L3)),aa_nat_int(semiri1314217659103216013at_int,N3)) ).

% int_sgnE
tff(fact_8652_div__eq__sgn__abs,axiom,
    ! [K2: int,L: int] :
      ( ( aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K2)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_eq_sgn_abs
tff(fact_8653_div__sgn__abs__cancel,axiom,
    ! [V2: int,K2: int,L: int] :
      ( ( V2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V2)),aa_int_int(abs_abs_int,K2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V2)),aa_int_int(abs_abs_int,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K2)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_8654_div__dvd__sgn__abs,axiom,
    ! [L: int,K2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_int_int(sgn_sgn_int,L))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K2)),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_8655_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K2: int,Q2: int] :
      ( ( aa_int_int(sgn_sgn_int,R) = 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,R)),aa_int_int(abs_abs_int,L)))
       => ( ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L)),R) )
         => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K2,L),product_Pair_int_int(Q2,R))) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_8656_eucl__rel__int_Osimps,axiom,
    ! [A12: int,A23: int,A32: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A12,A23),A32))
    <=> ( ? [K3: int] :
            ( ( A12 = K3 )
            & ( A23 = zero_zero_int )
            & ( A32 = product_Pair_int_int(zero_zero_int,K3) ) )
        | ? [L4: int,K3: int,Q5: int] :
            ( ( A12 = K3 )
            & ( A23 = L4 )
            & ( A32 = product_Pair_int_int(Q5,zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q5: int] :
            ( ( A12 = K3 )
            & ( A23 = L4 )
            & ( A32 = product_Pair_int_int(Q5,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,Q5),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_8657_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A32: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A12,A23),A32))
     => ( ( ( A23 = zero_zero_int )
         => ( A32 != product_Pair_int_int(zero_zero_int,A12) ) )
       => ( ! [Q3: int] :
              ( ( A32 = product_Pair_int_int(Q3,zero_zero_int) )
             => ( ( A23 != zero_zero_int )
               => ( A12 != aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A23) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = product_Pair_int_int(Q3,R3) )
               => ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A23) )
                 => ( 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,A23)))
                   => ( A12 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A23)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_8658_div__noneq__sgn__abs,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K2) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K2)),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),K2)))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_8659_divide__int__unfold,axiom,
    ! [L: int,K2: int,N: nat,M2: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K2) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_int ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K2) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)) ) )
          & ( ( aa_int_int(sgn_sgn_int,K2) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = 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,M2),N)),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M2)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_8660_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)) ).

% natLess_def
tff(fact_8661_modulo__int__def,axiom,
    ! [L: int,K2: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = K2 ) )
      & ( ( L != zero_zero_int )
       => ( ( ( aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K2) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),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),K2))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))))) ) ) ) ) ) ).

% modulo_int_def
tff(fact_8662_num_Osize__gen_I3_J,axiom,
    ! [X33: num] : aa_num_nat(size_num,aa_num_num(bit1,X33)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X33)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_8663_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% mask_nat_positive_iff
tff(fact_8664_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_8665_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( aa_int_nat(nat2,I2) = zero_zero_nat )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),zero_zero_int)) ) ).

% nat_0_iff
tff(fact_8666_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_8667_nat__neg__numeral,axiom,
    ! [K2: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_8668_nat__zminus__int,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_8669_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_Suc_0
tff(fact_8670_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_8671_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_8672_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_8673_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_8674_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: 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)),N)),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)),N)),A)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_8675_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: 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)),N)))
    <=> 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)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_8676_take__bit__tightened__less__eq__nat,axiom,
    ! [M2: nat,N: nat,Q2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M2),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),Q2))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_8677_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M2: 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,N),M2)),M2)) ).

% take_bit_nat_less_eq_self
tff(fact_8678_take__bit__mult,axiom,
    ! [N: nat,K2: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),K2)),aa_int_int(bit_se2923211474154528505it_int(N),L))) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,K2),L)) ).

% take_bit_mult
tff(fact_8679_less__eq__mask,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ).

% less_eq_mask
tff(fact_8680_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_8681_take__bit__tightened__less__eq__int,axiom,
    ! [M2: nat,N: nat,K2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(M2),K2)),aa_int_int(bit_se2923211474154528505it_int(N),K2))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_8682_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_8683_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% nat_le_iff
tff(fact_8684_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_8685_nat__times__as__int,axiom,
    ! [X5: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X5),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_times_as_int
tff(fact_8686_less__mask,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ) ).

% less_mask
tff(fact_8687_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_8688_nat__eq__iff2,axiom,
    ! [M2: nat,W2: int] :
      ( ( M2 = 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,M2) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( M2 = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
tff(fact_8689_nat__eq__iff,axiom,
    ! [W2: int,M2: nat] :
      ( ( aa_int_nat(nat2,W2) = M2 )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( W2 = aa_nat_int(semiri1314217659103216013at_int,M2) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( M2 = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
tff(fact_8690_split__nat,axiom,
    ! [P2: fun_nat_bool,I2: int] :
      ( pp(aa_nat_bool(P2,aa_int_nat(nat2,I2)))
    <=> ( ! [N4: nat] :
            ( ( I2 = aa_nat_int(semiri1314217659103216013at_int,N4) )
           => pp(aa_nat_bool(P2,N4)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),zero_zero_int))
         => pp(aa_nat_bool(P2,zero_zero_nat)) ) ) ) ).

% split_nat
tff(fact_8691_le__nat__iff,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_int_nat(nat2,K2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),K2)) ) ) ).

% le_nat_iff
tff(fact_8692_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_8693_Suc__as__int,axiom,
    ! [X5: nat] : aa_nat_nat(suc,X5) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X5)),one_one_int)) ).

% Suc_as_int
tff(fact_8694_nat__abs__triangle__ineq,axiom,
    ! [K2: 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,K2),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ).

% nat_abs_triangle_ineq
tff(fact_8695_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_8696_div__abs__eq__div__nat,axiom,
    ! [K2: int,L: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K2)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% div_abs_eq_div_nat
tff(fact_8697_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N ) ) ) ).

% floor_eq3
tff(fact_8698_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_8699_mod__abs__eq__div__nat,axiom,
    ! [K2: int,L: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(abs_abs_int,K2)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% mod_abs_eq_div_nat
tff(fact_8700_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_8701_sgn__power__injE,axiom,
    ! [A: real,N: nat,X: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N)) = X )
     => ( ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B)),N)) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_8702_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_8703_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_8704_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_8705_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N ) ) ) ).

% floor_eq4
tff(fact_8706_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_8707_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_8708_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M2)),M2))
    <=> 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))),N)),M2)) ) ).

% take_bit_nat_less_self_iff
tff(fact_8709_Suc__mask__eq__exp,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% Suc_mask_eq_exp
tff(fact_8710_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_8711_nat__dvd__iff,axiom,
    ! [Z2: int,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,Z2)),M2))
    <=> ( ( 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,M2))) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
         => ( M2 = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
tff(fact_8712_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_numeral_minus_bit0
tff(fact_8713_take__bit__int__less__eq,axiom,
    ! [N: nat,K2: 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))),N)),K2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(N),K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_8714_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K2: int] : aa_int_int(bit_ri631733984087533419it_int(N),K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_8715_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),X)) = aa_real_real(minus_minus_real(aa_real_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_8716_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : aa_num_nat(size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_8717_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K2))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_8718_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K2))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_8719_or__int__unfold,axiom,
    ! [K2: int,L: int] :
      ( ( ( ( K2 = aa_int_int(uminus_uminus_int,one_one_int) )
          | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L) = aa_int_int(uminus_uminus_int,one_one_int) ) )
      & ( ~ ( ( K2 = aa_int_int(uminus_uminus_int,one_one_int) )
            | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
       => ( ( ( K2 = zero_zero_int )
           => ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L) = L ) )
          & ( ( K2 != zero_zero_int )
           => ( ( ( L = zero_zero_int )
               => ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L) = K2 ) )
              & ( ( L != zero_zero_int )
               => ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),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(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% or_int_unfold
tff(fact_8720_arctan__half,axiom,
    ! [X: real] : aa_real_real(arctan,X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(divide_divide_real(X),aa_real_real(plus_plus_real(one_one_real),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_8721_real__sqrt__mult__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,A)),aa_real_real(sqrt,A)) = aa_real_real(abs_abs_real,A) ).

% real_sqrt_mult_self
tff(fact_8722_real__sqrt__abs2,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)) = aa_real_real(abs_abs_real,X) ).

% real_sqrt_abs2
tff(fact_8723_pred__numeral__inc,axiom,
    ! [K2: num] : aa_num_nat(pred_numeral,aa_num_num(inc,K2)) = aa_num_nat(numeral_numeral_nat,K2) ).

% pred_numeral_inc
tff(fact_8724_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_8725_real__sqrt__mult,axiom,
    ! [X: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)) ).

% real_sqrt_mult
tff(fact_8726_num__induct,axiom,
    ! [P2: fun_num_bool,X: num] :
      ( pp(aa_num_bool(P2,one))
     => ( ! [X3: num] :
            ( pp(aa_num_bool(P2,X3))
           => pp(aa_num_bool(P2,aa_num_num(inc,X3))) )
       => pp(aa_num_bool(P2,X)) ) ) ).

% num_induct
tff(fact_8727_add__inc,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,X),aa_num_num(inc,Y)) = aa_num_num(inc,aa_num_num(aa_num_fun_num_num(plus_plus_num,X),Y)) ).

% add_inc
tff(fact_8728_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_8729_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_8730_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_8731_add__One,axiom,
    ! [X: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,X),one) = aa_num_num(inc,X) ).

% add_One
tff(fact_8732_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))))) ).

% le_real_sqrt_sumsq
tff(fact_8733_inc__BitM__eq,axiom,
    ! [N: num] : aa_num_num(inc,aa_num_num(bitM,N)) = aa_num_num(bit0,N) ).

% inc_BitM_eq
tff(fact_8734_BitM__inc__eq,axiom,
    ! [N: num] : aa_num_num(bitM,aa_num_num(inc,N)) = aa_num_num(bit1,N) ).

% BitM_inc_eq
tff(fact_8735_mult__inc,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(times_times_num,X),aa_num_num(inc,Y)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(times_times_num,X),Y)),X) ).

% mult_inc
tff(fact_8736_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(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_8737_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(divide_divide_real(aa_real_real(plus_plus_real(X),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_8738_or__int__rec,axiom,
    ! [K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fdisj(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))),K2))),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(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_8739_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N: nat,K2: int] : aa_int_int(bit_ri631733984087533419it_int(N),K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),K2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),N)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_8740_cis__2pi,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_8741_take__bit__Suc__from__most,axiom,
    ! [N: nat,K2: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),N)))),aa_int_int(bit_se2923211474154528505it_int(N),K2)) ).

% take_bit_Suc_from_most
tff(fact_8742_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_wk)) ).

% pi_def
tff(fact_8743_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_8744_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_8745_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_8746_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_8747_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_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,N)))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W2))),N)) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_8748_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_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,N)))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W2)),N)) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_8749_cis__mult,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(plus_plus_real(A),B)) ).

% cis_mult
tff(fact_8750_DeMoivre,axiom,
    ! [A: real,N: nat] : aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),N) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),A)) ).

% DeMoivre
tff(fact_8751_bit__concat__bit__iff,axiom,
    ! [M2: nat,K2: int,L: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(M2),K2),L)),N))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
          & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),N)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
          & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_8752_int__bit__bound,axiom,
    ! [K2: int] :
      ~ ! [N3: nat] :
          ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N3),M3))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),M3))
              <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),N3)) ) )
         => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N3))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N3),one_one_nat)))
              <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),N3)) ) ) ) ).

% int_bit_bound
tff(fact_8753_set__bit__eq,axiom,
    ! [N: nat,K2: int] : aa_int_int(bit_se7879613467334960850it_int(N),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),N)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% set_bit_eq
tff(fact_8754_unset__bit__eq,axiom,
    ! [N: nat,K2: int] : aa_int_int(bit_se4203085406695923979it_int(N),K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% unset_bit_eq
tff(fact_8755_or__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),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))),N))) ).

% or_Suc_0_eq
tff(fact_8756_Suc__0__or__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),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))),N))) ).

% Suc_0_or_eq
tff(fact_8757_or__nat__rec,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M2),N) = 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))),M2))),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))),N))))),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,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_8758_or__nat__unfold,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M2),N) = N ) )
      & ( ( M2 != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M2),N) = M2 ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M2),N) = 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,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),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,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_8759_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_complex_bool(bij_betw_nat_complex(aTP_Lamm_wl(N),aa_nat_set_nat(set_ord_lessThan_nat,N)),collect_complex(aTP_Lamm_bl(N)))) ) ).

% bij_betw_roots_unity
tff(fact_8760_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( pp(aa_set_real_bool(member_real(N),ring_1_Ints_real))
     => ( aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),N)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_8761_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,N))) ).

% not_bit_Suc_0_Suc
tff(fact_8762_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),N))
    <=> ( N = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_8763_not__bit__Suc__0__numeral,axiom,
    ! [N: 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,N))) ).

% not_bit_Suc_0_numeral
tff(fact_8764_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 )
    <=> pp(aa_set_real_bool(member_real(X),ring_1_Ints_real)) ) ).

% sin_times_pi_eq_0
tff(fact_8765_sin__integer__2pi,axiom,
    ! [N: real] :
      ( pp(aa_set_real_bool(member_real(N),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)),N)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_8766_cos__integer__2pi,axiom,
    ! [N: real] :
      ( pp(aa_set_real_bool(member_real(N),ring_1_Ints_real))
     => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),N)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_8767_i__even__power,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) ).

% i_even_power
tff(fact_8768_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),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))),N)))),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))),N)))) ).

% Suc_0_xor_eq
tff(fact_8769_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),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,N),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))),N)))),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))),N)))) ).

% xor_Suc_0_eq
tff(fact_8770_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_8771_complex__i__mult__minus,axiom,
    ! [X: complex] : aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(imaginary_unit),X)) = aa_complex_complex(uminus1482373934393186551omplex,X) ).

% complex_i_mult_minus
tff(fact_8772_divide__numeral__i,axiom,
    ! [Z2: complex,N: num] : aa_complex_complex(divide1717551699836669952omplex(Z2),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(N)),imaginary_unit)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z2))),numera6690914467698888265omplex(N)) ).

% divide_numeral_i
tff(fact_8773_divide__i,axiom,
    ! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),imaginary_unit) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)),X) ).

% divide_i
tff(fact_8774_i__squared,axiom,
    aa_complex_complex(times_times_complex(imaginary_unit),imaginary_unit) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% i_squared
tff(fact_8775_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_8776_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_8777_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_8778_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_8779_i__times__eq__iff,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( aa_complex_complex(times_times_complex(imaginary_unit),W2) = Z2 )
    <=> ( W2 = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z2)) ) ) ).

% i_times_eq_iff
tff(fact_8780_i__mult__Complex,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B)),A) ).

% i_mult_Complex
tff(fact_8781_Complex__mult__i,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(A),B)),imaginary_unit) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B)),A) ).

% Complex_mult_i
tff(fact_8782_xor__nat__unfold,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M2),N) = N ) )
      & ( ( M2 != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M2),N) = M2 ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M2),N) = 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,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),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,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_8783_xor__nat__rec,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M2),N) = 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))),M2))),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))),N)))))),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,M2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_8784_Sum__Ico__nat,axiom,
    ! [M2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N)) = 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,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_8785_Least__eq__0,axiom,
    ! [P2: fun_nat_bool] :
      ( pp(aa_nat_bool(P2,zero_zero_nat))
     => ( ord_Least_nat(P2) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_8786_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_8787_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_8788_atLeastLessThan__singleton,axiom,
    ! [M2: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M2),aa_nat_nat(suc,M2)) = insert_nat(M2,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_8789_all__nat__less__eq,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( ! [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M6),N))
         => pp(aa_nat_bool(P2,M6)) )
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
         => pp(aa_nat_bool(P2,X4)) ) ) ).

% all_nat_less_eq
tff(fact_8790_ex__nat__less__eq,axiom,
    ! [N: nat,P2: fun_nat_bool] :
      ( ? [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M6),N))
          & pp(aa_nat_bool(P2,M6)) )
    <=> ? [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
          & pp(aa_nat_bool(P2,X4)) ) ) ).

% ex_nat_less_eq
tff(fact_8791_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_8792_lessThan__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N) ).

% lessThan_atLeast0
tff(fact_8793_atLeastLessThan0,axiom,
    ! [M2: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M2),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_8794_Least__Suc2,axiom,
    ! [P2: fun_nat_bool,N: nat,Q: fun_nat_bool,M2: nat] :
      ( pp(aa_nat_bool(P2,N))
     => ( pp(aa_nat_bool(Q,M2))
       => ( ~ pp(aa_nat_bool(P2,zero_zero_nat))
         => ( ! [K: nat] :
                ( pp(aa_nat_bool(P2,aa_nat_nat(suc,K)))
              <=> pp(aa_nat_bool(Q,K)) )
           => ( ord_Least_nat(P2) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_8795_Least__Suc,axiom,
    ! [P2: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P2,N))
     => ( ~ pp(aa_nat_bool(P2,zero_zero_nat))
       => ( ord_Least_nat(P2) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_wm(P2))) ) ) ) ).

% Least_Suc
tff(fact_8796_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(N,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ).

% atLeast0_lessThan_Suc
tff(fact_8797_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N5)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_8798_subset__card__intvl__is__intvl,axiom,
    ! [A4: set_nat,K2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A4),aa_nat_set_nat(set_or4665077453230672383an_nat(K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_set_nat_nat(finite_card_nat,A4)))))
     => ( A4 = aa_nat_set_nat(set_or4665077453230672383an_nat(K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_set_nat_nat(finite_card_nat,A4))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_8799_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(I2),J2)),aa_nat_set_nat(set_or4665077453230672383an_nat(J2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_8800_atLeastLessThanSuc,axiom,
    ! [M2: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M2),aa_nat_nat(suc,N)) = insert_nat(N,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M2),aa_nat_nat(suc,N)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
tff(fact_8801_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_Suc_fact
tff(fact_8802_prod__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_fact
tff(fact_8803_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N5: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N5)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_8804_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_ar),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_set_nat_nat(finite_card_nat,S3)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),S3))) ).

% card_sum_le_nat_sum
tff(fact_8805_atLeastLessThan__nat__numeral,axiom,
    ! [M2: nat,K2: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_num_nat(pred_numeral,K2)))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M2),aa_num_nat(numeral_numeral_nat,K2)) = insert_nat(aa_num_nat(pred_numeral,K2),aa_nat_set_nat(set_or4665077453230672383an_nat(M2),aa_num_nat(pred_numeral,K2))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_num_nat(pred_numeral,K2)))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M2),aa_num_nat(numeral_numeral_nat,K2)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_8806_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,N)),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_8807_xor__int__rec,axiom,
    ! [K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,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))),K2))),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(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_8808_sum__power2,axiom,
    ! [K2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)),one_one_nat) ).

% sum_power2
tff(fact_8809_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(A,I3)),aa_nat_nat(A,J3))) ) )
     => ( ! [I3: nat,J3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J3))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(B,J3)),aa_nat_nat(B,I3))) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_lg(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))),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),N))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_8810_xor__int__unfold,axiom,
    ! [K2: int,L: int] :
      ( ( ( K2 = aa_int_int(uminus_uminus_int,one_one_int) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L) = aa_int_int(bit_ri7919022796975470100ot_int,L) ) )
      & ( ( K2 != aa_int_int(uminus_uminus_int,one_one_int) )
       => ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L) = aa_int_int(bit_ri7919022796975470100ot_int,K2) ) )
          & ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
           => ( ( ( K2 = zero_zero_int )
               => ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L) = L ) )
              & ( ( K2 != zero_zero_int )
               => ( ( ( L = zero_zero_int )
                   => ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L) = K2 ) )
                  & ( ( L != zero_zero_int )
                   => ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),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(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
tff(fact_8811_exp__two__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_8812_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_8813_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_8814_exp__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,pi))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i'
tff(fact_8815_exp__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,pi)),imaginary_unit)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i
tff(fact_8816_exp__two__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_real_complex(real_V4546457046886955230omplex,pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_8817_complex__exp__exists,axiom,
    ! [Z2: complex] :
    ? [A3: complex,R3: real] : Z2 = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(exp_complex,A3)) ).

% complex_exp_exists
tff(fact_8818_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_8819_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_8820_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y: real,R: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(X),Y)),aa_real_complex(real_V4546457046886955230omplex,R)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),R)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),R)) ).

% Complex_mult_complex_of_real
tff(fact_8821_complex__of__real__mult__Complex,axiom,
    ! [R: real,X: real,Y: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R)),aa_real_complex(complex2(X),Y)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),Y)) ).

% complex_of_real_mult_Complex
tff(fact_8822_Complex__eq,axiom,
    ! [A: real,B: real] : aa_real_complex(complex2(A),B) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,A)),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% Complex_eq
tff(fact_8823_cis__conv__exp,axiom,
    ! [B: real] : aa_real_complex(cis,B) = aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% cis_conv_exp
tff(fact_8824_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_8825_complex__of__real__i,axiom,
    ! [R: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R)),imaginary_unit) = aa_real_complex(complex2(zero_zero_real),R) ).

% complex_of_real_i
tff(fact_8826_i__complex__of__real,axiom,
    ! [R: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,R)) = aa_real_complex(complex2(zero_zero_real),R) ).

% i_complex_of_real
tff(fact_8827_complex__split__polar,axiom,
    ! [Z2: complex] :
    ? [R3: real,A3: real] : Z2 = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A3))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A3))))) ).

% complex_split_polar
tff(fact_8828_and__not__numerals_I5_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M2))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% and_not_numerals(5)
tff(fact_8829_and__not__numerals_I6_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M2))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% and_not_numerals(6)
tff(fact_8830_and__not__numerals_I9_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M2))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% and_not_numerals(9)
tff(fact_8831_or__not__numerals_I6_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M2))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% or_not_numerals(6)
tff(fact_8832_cmod__unit__one,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A))))) = one_one_real ).

% cmod_unit_one
tff(fact_8833_cmod__complex__polar,axiom,
    ! [R: real,A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A)))))) = aa_real_real(abs_abs_real,R) ).

% cmod_complex_polar
tff(fact_8834_or__not__numerals_I5_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M2))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(5)
tff(fact_8835_and__not__numerals_I8_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M2))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% and_not_numerals(8)
tff(fact_8836_or__not__numerals_I8_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M2))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(8)
tff(fact_8837_or__not__numerals_I9_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M2))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(9)
tff(fact_8838_not__int__rec,axiom,
    ! [K2: int] : aa_int_int(bit_ri7919022796975470100ot_int,K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_8839_Cauchy__iff2,axiom,
    ! [X6: fun_nat_real] :
      ( topolo4055970368930404560y_real(X6)
    <=> ! [J: 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(minus_minus_real(aa_nat_real(X6,M6)),aa_nat_real(X6,N4)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J))))) ) ) ) ).

% Cauchy_iff2
tff(fact_8840_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_8841_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_8842_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_wn(L),Qr) ).

% divmod_step_integer_def
tff(fact_8843_complex__div__cnj,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))),aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_div_cnj
tff(fact_8844_list__decode_Opsimps_I2_J,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N)))
     => ( aa_nat_list_nat(nat_list_decode,aa_nat_nat(suc,N)) = produc2761476792215241774st_nat(aTP_Lamm_wo,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ) ) ).

% list_decode.psimps(2)
tff(fact_8845_complex__cnj__mult,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(times_times_complex(X),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(cnj,X)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_mult
tff(fact_8846_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_8847_times__integer__code_I1_J,axiom,
    ! [K2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% times_integer_code(1)
tff(fact_8848_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_8849_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_8850_list__decode_Osimps_I2_J,axiom,
    ! [N: nat] : aa_nat_list_nat(nat_list_decode,aa_nat_nat(suc,N)) = produc2761476792215241774st_nat(aTP_Lamm_wo,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ).

% list_decode.simps(2)
tff(fact_8851_complex__mod__mult__cnj,axiom,
    ! [Z2: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Z2),aa_complex_complex(cnj,Z2))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% complex_mod_mult_cnj
tff(fact_8852_complex__norm__square,axiom,
    ! [Z2: complex] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(Z2),aa_complex_complex(cnj,Z2)) ).

% complex_norm_square
tff(fact_8853_integer__of__int__code,axiom,
    ! [K2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
       => ( aa_int_Code_integer(code_integer_of_int,K2) = uminus1351360451143612070nteger(aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K2))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K2),zero_zero_int))
       => ( ( ( K2 = zero_zero_int )
           => ( aa_int_Code_integer(code_integer_of_int,K2) = zero_z3403309356797280102nteger ) )
          & ( ( K2 != zero_zero_int )
           => ( aa_int_Code_integer(code_integer_of_int,K2) = if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),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(aa_int_fun_int_int(divide_divide_int,K2),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(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_Code_integer)) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_8854_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)) ) )
         => ~ ! [N3: nat] :
                ( ( X = aa_nat_nat(suc,N3) )
               => ( ( Y = produc2761476792215241774st_nat(aTP_Lamm_wo,aa_nat7603221163087113075at_nat(nat_prod_decode,N3)) )
                 => ~ pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N3))) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_8855_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 ) )
       => ~ ! [N3: nat] :
              ( ( X = aa_nat_nat(suc,N3) )
             => ( Y != produc2761476792215241774st_nat(aTP_Lamm_wo,aa_nat7603221163087113075at_nat(nat_prod_decode,N3)) ) ) ) ) ).

% list_decode.elims
tff(fact_8856_cnj__add__mult__eq__Re,axiom,
    ! [Z2: complex,W2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Z2),aa_complex_complex(cnj,W2))),aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Z2)),W2)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,aa_complex_complex(times_times_complex(Z2),aa_complex_complex(cnj,W2))))) ).

% cnj_add_mult_eq_Re
tff(fact_8857_list__encode_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X3: nat,Xs2: list_nat] : X != cons_nat(X3,Xs2) ) ).

% list_encode.cases
tff(fact_8858_times__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,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(times_times_int,Xa2),X)) ).

% times_integer.abs_eq
tff(fact_8859_scaleR__complex_Osimps_I1_J,axiom,
    ! [R: real,X: complex] : aa_complex_real(re,aa_complex_complex(real_V2046097035970521341omplex(R),X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_complex_real(re,X)) ).

% scaleR_complex.simps(1)
tff(fact_8860_list__decode_Osimps_I1_J,axiom,
    aa_nat_list_nat(nat_list_decode,zero_zero_nat) = nil_nat ).

% list_decode.simps(1)
tff(fact_8861_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_8862_complex__mod__sqrt__Re__mult__cnj,axiom,
    ! [Z2: complex] : aa_complex_real(real_V1022390504157884413omplex,Z2) = aa_real_real(sqrt,aa_complex_real(re,aa_complex_complex(times_times_complex(Z2),aa_complex_complex(cnj,Z2)))) ).

% complex_mod_sqrt_Re_mult_cnj
tff(fact_8863_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_8864_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_8865_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_8866_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_8867_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_8868_cos__n__Re__cis__pow__n,axiom,
    ! [N: nat,A: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),A)) = aa_complex_real(re,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),N)) ).

% cos_n_Re_cis_pow_n
tff(fact_8869_complex__add__cnj,axiom,
    ! [Z2: complex] : aa_complex_complex(plus_plus_complex(Z2),aa_complex_complex(cnj,Z2)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,Z2))) ).

% complex_add_cnj
tff(fact_8870_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 ) )
       => ~ ! [X3: nat,Xs2: list_nat] :
              ( ( X = cons_nat(X3,Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),aa_list_nat_nat(nat_list_encode,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_8871_Complex__divide,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_real_complex(complex2(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% Complex_divide
tff(fact_8872_Re__Reals__divide,axiom,
    ! [R: complex,Z2: complex] :
      ( pp(aa_set_complex_bool(member_complex(R),real_V2521375963428798218omplex))
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(R),Z2)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,R)),aa_complex_real(re,Z2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Re_Reals_divide
tff(fact_8873_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_8874_list__decode__inverse,axiom,
    ! [N: nat] : aa_list_nat_nat(nat_list_encode,aa_nat_list_nat(nat_list_decode,N)) = N ).

% list_decode_inverse
tff(fact_8875_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_8876_Im__i__times,axiom,
    ! [Z2: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(imaginary_unit),Z2)) = aa_complex_real(re,Z2) ).

% Im_i_times
tff(fact_8877_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_8878_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_8879_Re__i__times,axiom,
    ! [Z2: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(imaginary_unit),Z2)) = aa_real_real(uminus_uminus_real,aa_complex_real(im,Z2)) ).

% Re_i_times
tff(fact_8880_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_8881_scaleR__complex_Osimps_I2_J,axiom,
    ! [R: real,X: complex] : aa_complex_real(im,aa_complex_complex(real_V2046097035970521341omplex(R),X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_complex_real(im,X)) ).

% scaleR_complex.simps(2)
tff(fact_8882_list__encode_Osimps_I1_J,axiom,
    aa_list_nat_nat(nat_list_encode,nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_8883_times__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(X),Y)) = aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(re,Y))) ).

% times_complex.simps(2)
tff(fact_8884_times__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(X),Y)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y))) ).

% times_complex.simps(1)
tff(fact_8885_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_8886_scaleR__complex_Ocode,axiom,
    ! [R: real,X: complex] : aa_complex_complex(real_V2046097035970521341omplex(R),X) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_complex_real(re,X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_complex_real(im,X))) ).

% scaleR_complex.code
tff(fact_8887_Im__Reals__divide,axiom,
    ! [R: complex,Z2: complex] :
      ( pp(aa_set_complex_bool(member_complex(R),real_V2521375963428798218omplex))
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(R),Z2)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_complex_real(re,R))),aa_complex_real(im,Z2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Im_Reals_divide
tff(fact_8888_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_8889_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_8890_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_8891_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_8892_sin__n__Im__cis__pow__n,axiom,
    ! [N: nat,A: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),A)) = aa_complex_real(im,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),N)) ).

% sin_n_Im_cis_pow_n
tff(fact_8893_Re__exp,axiom,
    ! [Z2: complex] : aa_complex_real(re,aa_complex_complex(exp_complex,Z2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z2))),aa_real_real(cos_real,aa_complex_real(im,Z2))) ).

% Re_exp
tff(fact_8894_Im__exp,axiom,
    ! [Z2: complex] : aa_complex_real(im,aa_complex_complex(exp_complex,Z2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z2))),aa_real_real(sin_real,aa_complex_real(im,Z2))) ).

% Im_exp
tff(fact_8895_complex__eq,axiom,
    ! [A: complex] : A = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(re,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(im,A)))) ).

% complex_eq
tff(fact_8896_times__complex_Ocode,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(times_times_complex(X),Y) = aa_real_complex(complex2(aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y)))),aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(re,Y)))) ).

% times_complex.code
tff(fact_8897_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_8898_exp__eq__polar,axiom,
    ! [Z2: complex] : aa_complex_complex(exp_complex,Z2) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(exp_real,aa_complex_real(re,Z2)))),aa_real_complex(cis,aa_complex_real(im,Z2))) ).

% exp_eq_polar
tff(fact_8899_Im__power2,axiom,
    ! [X: complex] : aa_complex_real(im,aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,X))),aa_complex_real(im,X)) ).

% Im_power2
tff(fact_8900_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_8901_complex__diff__cnj,axiom,
    ! [Z2: complex] : aa_complex_complex(minus_minus_complex(Z2),aa_complex_complex(cnj,Z2)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(im,Z2)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_8902_Re__divide,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Re_divide
tff(fact_8903_complex__mult__cnj,axiom,
    ! [Z2: complex] : aa_complex_complex(times_times_complex(Z2),aa_complex_complex(cnj,Z2)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_mult_cnj
tff(fact_8904_Im__divide,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Im_divide
tff(fact_8905_complex__abs__le__norm,axiom,
    ! [Z2: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,aa_complex_real(re,Z2))),aa_real_real(abs_abs_real,aa_complex_real(im,Z2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_complex_real(real_V1022390504157884413omplex,Z2)))) ).

% complex_abs_le_norm
tff(fact_8906_csqrt_Ocode,axiom,
    ! [Z2: complex] : aa_complex_complex(csqrt,Z2) = aa_real_complex(complex2(aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,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(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(divide_divide_real(aa_real_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_8907_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(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(divide_divide_real(aa_real_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_8908_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_8909_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_8910_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = product_Pair_int_int(one_one_int,zero_zero_int) ).

% bezw_0
tff(fact_8911_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),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))),N) ).

% push_bit_of_Suc_0
tff(fact_8912_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fTrue,aTP_Lamm_wp),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_8913_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aTP_Lamm_wq),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_8914_bit__push__bit__iff__int,axiom,
    ! [M2: nat,K2: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_se545348938243370406it_int(M2),K2)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).

% bit_push_bit_iff_int
tff(fact_8915_bit__push__bit__iff__nat,axiom,
    ! [M2: nat,Q2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M2),Q2)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_8916_less__eq__nat_Osimps_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N))
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aa_nat_fun_nat_bool(ord_less_eq_nat,M2)),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_8917_max__Suc2,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M2),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_wr,N)),M2) ).

% max_Suc2
tff(fact_8918_max__Suc1,axiom,
    ! [N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),M2) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_ws,N)),M2) ).

% max_Suc1
tff(fact_8919_diff__Suc,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_ar),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) ).

% diff_Suc
tff(fact_8920_push__bit__nat__def,axiom,
    ! [N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% push_bit_nat_def
tff(fact_8921_push__bit__int__def,axiom,
    ! [N: nat,K2: int] : aa_int_int(bit_se545348938243370406it_int(N),K2) = aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% push_bit_int_def
tff(fact_8922_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_ar),Nat) ).

% pred_def
tff(fact_8923_Suc__0__mod__numeral,axiom,
    ! [K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K2)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K2)) ).

% Suc_0_mod_numeral
tff(fact_8924_Suc__0__div__numeral,axiom,
    ! [K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K2)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K2)) ).

% Suc_0_div_numeral
tff(fact_8925_nat__of__integer__code,axiom,
    ! [K2: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K2),zero_z3403309356797280102nteger))
       => ( aa_Code_integer_nat(code_nat_of_integer,K2) = zero_zero_nat ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K2),zero_z3403309356797280102nteger))
       => ( aa_Code_integer_nat(code_nat_of_integer,K2) = produc1555791787009142072er_nat(aTP_Lamm_wt,code_divmod_integer(K2,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% nat_of_integer_code
tff(fact_8926_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% drop_bit_of_Suc_0
tff(fact_8927_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K2: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(bit_se8568078237143864401it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) ).

% drop_bit_Suc_minus_bit0
tff(fact_8928_fst__divmod__nat,axiom,
    ! [M2: nat,N: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,divmod_nat(M2,N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) ).

% fst_divmod_nat
tff(fact_8929_nat__of__integer__non__positive,axiom,
    ! [K2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K2),zero_z3403309356797280102nteger))
     => ( aa_Code_integer_nat(code_nat_of_integer,K2) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_8930_snd__divmod__nat,axiom,
    ! [M2: nat,N: nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,divmod_nat(M2,N)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N) ).

% snd_divmod_nat
tff(fact_8931_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K2: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(bit_se8568078237143864401it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K2)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_8932_nat__of__integer__code__post_I1_J,axiom,
    aa_Code_integer_nat(code_nat_of_integer,zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_8933_int__of__integer__code,axiom,
    ! [K2: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger))
       => ( aa_Code_integer_int(code_int_of_integer,K2) = aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,uminus1351360451143612070nteger(K2))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger))
       => ( ( ( K2 = zero_z3403309356797280102nteger )
           => ( aa_Code_integer_int(code_int_of_integer,K2) = zero_zero_int ) )
          & ( ( K2 != zero_z3403309356797280102nteger )
           => ( aa_Code_integer_int(code_int_of_integer,K2) = produc1553301316500091796er_int(aTP_Lamm_wu,code_divmod_integer(K2,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ) ) ).

% int_of_integer_code
tff(fact_8934_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Xa2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa2)) ).

% times_integer.rep_eq
tff(fact_8935_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_8936_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_8937_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_8938_one__mod__minus__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,N)),product_snd_int_int(unique5052692396658037445od_int(one,N)))) ).

% one_mod_minus_numeral
tff(fact_8939_minus__one__mod__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,N)),product_snd_int_int(unique5052692396658037445od_int(one,N))) ).

% minus_one_mod_numeral
tff(fact_8940_numeral__mod__minus__numeral,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,N)),product_snd_int_int(unique5052692396658037445od_int(M2,N)))) ).

% numeral_mod_minus_numeral
tff(fact_8941_minus__numeral__mod__numeral,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M2))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,N)),product_snd_int_int(unique5052692396658037445od_int(M2,N))) ).

% minus_numeral_mod_numeral
tff(fact_8942_Divides_Oadjust__mod__def,axiom,
    ! [R: int,L: int] :
      ( ( ( R = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(adjust_mod,L),R) = zero_zero_int ) )
      & ( ( R != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(adjust_mod,L),R) = aa_int_int(aa_int_fun_int_int(minus_minus_int,L),R) ) ) ) ).

% Divides.adjust_mod_def
tff(fact_8943_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_8944_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_8945_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_8946_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_8947_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_8948_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_8949_Suc__funpow,axiom,
    ! [N: nat] : compow_fun_nat_nat(N,suc) = aa_nat_fun_nat_nat(plus_plus_nat,N) ).

% Suc_funpow
tff(fact_8950_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_8951_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_8952_sub__BitM__One__eq,axiom,
    ! [N: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bitM,N)),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(N),one)) ).

% sub_BitM_One_eq
tff(fact_8953_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => 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,N),aa_complex_real(real_V1022390504157884413omplex,C2)))),aa_real_complex(cis,aa_real_real(divide_divide_real(aa_complex_real(arg,C2)),aa_nat_real(semiri5074537144036343181t_real,N))))),collect_complex(aTP_Lamm_bl(N))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_bm(C2),N)))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_8954_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_ah,aTP_Lamm_hm) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_8955_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_8956_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(aa_nat_fun_real_real(root,N),Y) )
      <=> ( X = Y ) ) ) ).

% real_root_eq_iff
tff(fact_8957_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_8958_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_8959_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% real_root_less_iff
tff(fact_8960_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% real_root_le_iff
tff(fact_8961_real__root__one,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_8962_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = one_one_real )
      <=> ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_8963_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) ) ).

% real_root_gt_0_iff
tff(fact_8964_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),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_8965_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y)) ) ) ).

% real_root_ge_0_iff
tff(fact_8966_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),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_8967_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y)) ) ) ).

% real_root_gt_1_iff
tff(fact_8968_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),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_8969_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y)) ) ) ).

% real_root_ge_1_iff
tff(fact_8970_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),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_8971_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_8972_real__root__mult__exp,axiom,
    ! [M2: nat,N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)),X) = aa_real_real(aa_nat_fun_real_real(root,M2),aa_real_real(aa_nat_fun_real_real(root,N),X)) ).

% real_root_mult_exp
tff(fact_8973_real__root__mult,axiom,
    ! [N: nat,X: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)) ).

% real_root_mult
tff(fact_8974_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).

% real_root_less_mono
tff(fact_8975_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).

% real_root_le_mono
tff(fact_8976_real__root__power,axiom,
    ! [N: nat,X: real,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),K2)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),K2) ) ) ).

% real_root_power
tff(fact_8977_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(abs_abs_real,X)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) ) ) ).

% real_root_abs
tff(fact_8978_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) = aa_real_real(sgn_sgn_real,X) ) ) ).

% sgn_root
tff(fact_8979_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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,N),X))) ) ) ).

% real_root_gt_zero
tff(fact_8980_real__root__strict__decreasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( 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(ord_less_nat,N),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,N),X))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_8981_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_8982_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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,N),X))) ) ) ).

% real_root_pos_pos
tff(fact_8983_real__root__strict__increasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( 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(ord_less_nat,N),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,N),X)),aa_real_real(aa_nat_fun_real_real(root,N5),X))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_8984_real__root__decreasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( 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(ord_less_eq_nat,N),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,N),X))) ) ) ) ).

% real_root_decreasing
tff(fact_8985_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ) ).

% real_root_pow_pos
tff(fact_8986_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N) = X )
         => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_8987_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_8988_real__root__increasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( 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(ord_less_eq_nat,N),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,N),X)),aa_real_real(aa_nat_fun_real_real(root,N5),X))) ) ) ) ) ).

% real_root_increasing
tff(fact_8989_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X))),N)) = X ) ) ).

% sgn_power_root
tff(fact_8990_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y)),N))) = Y ) ) ).

% root_sgn_power
tff(fact_8991_ln__root,axiom,
    ! [N: nat,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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,N),B)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% ln_root
tff(fact_8992_log__root,axiom,
    ! [N: nat,A: real,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ( aa_real_real(log2(B),aa_real_real(aa_nat_fun_real_real(root,N),A)) = aa_real_real(divide_divide_real(aa_real_real(log2(B),A)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% log_root
tff(fact_8993_log__base__root,axiom,
    ! [N: nat,B: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => ( aa_real_real(log2(aa_real_real(aa_nat_fun_real_real(root,N),B)),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log2(B),X)) ) ) ) ).

% log_base_root
tff(fact_8994_split__root,axiom,
    ! [P2: fun_real_bool,N: nat,X: real] :
      ( pp(aa_real_bool(P2,aa_real_real(aa_nat_fun_real_real(root,N),X)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_real_bool(P2,zero_zero_real)) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ! [Y5: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y5)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y5)),N)) = X )
             => pp(aa_real_bool(P2,Y5)) ) ) ) ) ).

% split_root
tff(fact_8995_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(powr_real(X),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% root_powr_inverse
tff(fact_8996_powr__powr,axiom,
    ! [X: real,A: real,B: real] : aa_real_real(powr_real(aa_real_real(powr_real(X),A)),B) = aa_real_real(powr_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% powr_powr
tff(fact_8997_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(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(powr_real(X),A)),aa_real_real(powr_real(Y),A)) ) ) ) ).

% powr_mult
tff(fact_8998_divide__powr__uminus,axiom,
    ! [A: real,B: real,C2: real] : aa_real_real(divide_divide_real(A),aa_real_real(powr_real(B),C2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(powr_real(B),aa_real_real(uminus_uminus_real,C2))) ).

% divide_powr_uminus
tff(fact_8999_ln__powr,axiom,
    ! [X: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_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_9000_log__powr,axiom,
    ! [X: real,B: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(log2(B),aa_real_real(powr_real(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(log2(B),X)) ) ) ).

% log_powr
tff(fact_9001_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(powr_real(X),Y)) = aa_real_real(powr_real(X),aa_real_real(plus_plus_real(one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_9002_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(powr_real(aa_real_real(ln_ln_real,X)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(powr_real(A),A)),X))) ) ) ).

% ln_powr_bound2
tff(fact_9003_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(plus_plus_real(aa_real_real(log2(B),X)),Y) = aa_real_real(log2(B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(powr_real(B),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_9004_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(plus_plus_real(Y),aa_real_real(log2(B),X)) = aa_real_real(log2(B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(powr_real(B),Y)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_9005_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(minus_minus_real(aa_real_real(log2(B),X)),Y) = aa_real_real(log2(B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(powr_real(B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_9006_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_ww),Xa2),X)) ).

% times_int.abs_eq
tff(fact_9007_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = aa_num_num(inc,aa_nat_num(num_of_nat,N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = one ) ) ) ).

% num_of_nat.simps(2)
tff(fact_9008_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q2)) = Q2 ).

% num_of_nat_numeral_eq
tff(fact_9009_eq__Abs__Integ,axiom,
    ! [Z2: int] :
      ~ ! [X3: nat,Y3: nat] : Z2 != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3)) ).

% eq_Abs_Integ
tff(fact_9010_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9011_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_9012_int__def,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),zero_zero_nat)) ).

% int_def
tff(fact_9013_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_9014_num__of__nat__One,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),one_one_nat))
     => ( aa_nat_num(num_of_nat,N) = one ) ) ).

% num_of_nat_One
tff(fact_9015_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_wx),X)) ).

% uminus_int.abs_eq
tff(fact_9016_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_9017_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)) = aa_num_num(bit0,aa_nat_num(num_of_nat,N)) ) ) ).

% num_of_nat_double
tff(fact_9018_num__of__nat__plus__distrib,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,M2)),aa_nat_num(num_of_nat,N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9019_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_wz),Xa2),X)) ) ).

% less_eq_int.abs_eq
tff(fact_9020_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_xb),Xa2),X)) ).

% plus_int.abs_eq
tff(fact_9021_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_xd),Xa2),X)) ).

% minus_int.abs_eq
tff(fact_9022_sorted__list__of__set__atMost__Suc,axiom,
    ! [K2: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K2))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K2)),cons_nat(aa_nat_nat(suc,K2),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_9023_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K2: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K2))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K2)),cons_nat(K2,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_9024_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I2)),J2))
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I2),J2)) = cons_nat(aa_nat_nat(suc,I2),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I2)),J2))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9025_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J2: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),aa_nat_nat(suc,I2))))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I2),J2))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9026_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_wz),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).

% less_eq_int.rep_eq
tff(fact_9027_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),Y))
       => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xe,C2),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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C2)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C2),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_xe,C2),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_xe,C2),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9028_bij__betw__Suc,axiom,
    ! [M5: set_nat,N5: set_nat] :
      ( pp(aa_set_nat_bool(bij_betw_nat_nat(suc,M5),N5))
    <=> ( image_nat_nat(suc,M5) = N5 ) ) ).

% bij_betw_Suc
tff(fact_9029_image__Suc__atLeastAtMost,axiom,
    ! [I2: nat,J2: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(I2),J2)) = aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,I2)),aa_nat_nat(suc,J2)) ).

% image_Suc_atLeastAtMost
tff(fact_9030_image__Suc__atLeastLessThan,axiom,
    ! [I2: nat,J2: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(I2),J2)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,I2)),aa_nat_nat(suc,J2)) ).

% image_Suc_atLeastLessThan
tff(fact_9031_zero__notin__Suc__image,axiom,
    ! [A4: set_nat] : ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),image_nat_nat(suc,A4))) ).

% zero_notin_Suc_image
tff(fact_9032_image__Suc__lessThan,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N) ).

% image_Suc_lessThan
tff(fact_9033_image__Suc__atMost,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,N)) ).

% image_Suc_atMost
tff(fact_9034_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9035_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9036_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9037_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% atMost_Suc_eq_insert_0
tff(fact_9038_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J2: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I2)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I2),J2))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9039_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_wx)) ).

% uminus_int_def
tff(fact_9040_rat__inverse__code,axiom,
    ! [P: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P)) = produc4245557441103728435nt_int(aTP_Lamm_xf,quotient_of(P)) ).

% rat_inverse_code
tff(fact_9041_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_9042_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_9043_divide__rat__def,axiom,
    ! [Q2: rat,R: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Q2),R) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Q2),aa_rat_rat(inverse_inverse_rat,R)) ).

% divide_rat_def
tff(fact_9044_rat__less__eq__code,axiom,
    ! [P: rat,Q2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,P),Q2))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_xh(Q2)),quotient_of(P))) ) ).

% rat_less_eq_code
tff(fact_9045_rat__less__code,axiom,
    ! [P: rat,Q2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P),Q2))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_xj(Q2)),quotient_of(P))) ) ).

% rat_less_code
tff(fact_9046_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_9047_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_9048_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_9049_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,I2)),J2))
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I2),J2)) = cons_nat(aa_nat_nat(suc,I2),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I2)),J2))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9050_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_xk,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_9051_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_9052_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_ww)) ).

% times_int_def
tff(fact_9053_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_xd)) ).

% minus_int_def
tff(fact_9054_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_xb)) ).

% plus_int_def
tff(fact_9055_rat__minus__code,axiom,
    ! [P: rat,Q2: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,P),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_xm(Q2),quotient_of(P)) ).

% rat_minus_code
tff(fact_9056_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_9057_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_9058_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_9059_normalize__crossproduct,axiom,
    ! [Q2: int,S2: int,P: int,R: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S2 != zero_zero_int )
       => ( ( normalize(product_Pair_int_int(P,Q2)) = normalize(product_Pair_int_int(R,S2)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P),S2) = aa_int_int(aa_int_fun_int_int(times_times_int,R),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9060_rat__times__code,axiom,
    ! [P: rat,Q2: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_xo(Q2),quotient_of(P)) ).

% rat_times_code
tff(fact_9061_rat__divide__code,axiom,
    ! [P: rat,Q2: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_xq(Q2),quotient_of(P)) ).

% rat_divide_code
tff(fact_9062_rat__plus__code,axiom,
    ! [P: rat,Q2: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_xs(Q2),quotient_of(P)) ).

% rat_plus_code
tff(fact_9063_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)) ) )
         => ~ ! [X3: nat,Xs2: list_nat] :
                ( ( X = cons_nat(X3,Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),aa_list_nat_nat(nat_list_encode,Xs2)))) )
                 => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),cons_nat(X3,Xs2))) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_9064_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M2),aa_num_nat(numeral_numeral_nat,N)) = cons_nat(aa_num_nat(numeral_numeral_nat,M2),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M2)),aa_num_nat(numeral_numeral_nat,N)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M2),aa_num_nat(numeral_numeral_nat,N)) = nil_nat ) ) ) ).

% upt_rec_numeral
tff(fact_9065_tl__upt,axiom,
    ! [M2: nat,N: nat] : tl_nat(upt(M2,N)) = upt(aa_nat_nat(suc,M2),N) ).

% tl_upt
tff(fact_9066_length__upt,axiom,
    ! [I2: nat,J2: nat] : aa_list_nat_nat(size_size_list_nat,upt(I2,J2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I2) ).

% length_upt
tff(fact_9067_take__upt,axiom,
    ! [I2: nat,M2: nat,N: 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,I2),M2)),N))
     => ( take_nat(M2,upt(I2,N)) = upt(I2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),M2)) ) ) ).

% take_upt
tff(fact_9068_upt__conv__Nil,axiom,
    ! [J2: nat,I2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),I2))
     => ( upt(I2,J2) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_9069_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J2: nat] :
      ( ( upt(I2,J2) = nil_nat )
    <=> ( ( J2 = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),I2)) ) ) ).

% upt_eq_Nil_conv
tff(fact_9070_atLeastAtMost__upt,axiom,
    ! [N: nat,M2: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(N),M2) = set_nat2(upt(N,aa_nat_nat(suc,M2))) ).

% atLeastAtMost_upt
tff(fact_9071_atLeast__upt,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = set_nat2(upt(zero_zero_nat,N)) ).

% atLeast_upt
tff(fact_9072_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list_nat,Q2: nat] :
      ( ( cons_nat(M2,cons_nat(N,Ns)) = upt(M2,Q2) )
    <=> ( cons_nat(N,Ns) = upt(aa_nat_nat(suc,M2),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_9073_upt__0,axiom,
    ! [I2: nat] : upt(I2,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_9074_greaterThanAtMost__upt,axiom,
    ! [N: nat,M2: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(N),M2) = set_nat2(upt(aa_nat_nat(suc,N),aa_nat_nat(suc,M2))) ).

% greaterThanAtMost_upt
tff(fact_9075_greaterThanLessThan__upt,axiom,
    ! [N: nat,M2: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(N),M2) = set_nat2(upt(aa_nat_nat(suc,N),M2)) ).

% greaterThanLessThan_upt
tff(fact_9076_atMost__upto,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,N))) ).

% atMost_upto
tff(fact_9077_upt__conv__Cons,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
     => ( upt(I2,J2) = cons_nat(I2,upt(aa_nat_nat(suc,I2),J2)) ) ) ).

% upt_conv_Cons
tff(fact_9078_upt__add__eq__append,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
     => ( upt(I2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2)) = append_nat(upt(I2,J2),upt(J2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2))) ) ) ).

% upt_add_eq_append
tff(fact_9079_upt__rec,axiom,
    ! [I2: nat,J2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
       => ( upt(I2,J2) = cons_nat(I2,upt(aa_nat_nat(suc,I2),J2)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
       => ( upt(I2,J2) = nil_nat ) ) ) ).

% upt_rec
tff(fact_9080_upt__Suc,axiom,
    ! [I2: nat,J2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
       => ( upt(I2,aa_nat_nat(suc,J2)) = append_nat(upt(I2,J2),cons_nat(J2,nil_nat)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
       => ( upt(I2,aa_nat_nat(suc,J2)) = nil_nat ) ) ) ).

% upt_Suc
tff(fact_9081_upt__Suc__append,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
     => ( upt(I2,aa_nat_nat(suc,J2)) = append_nat(upt(I2,J2),cons_nat(J2,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_9082_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_9083_mono__times__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N)) ) ).

% mono_times_nat
tff(fact_9084_mono__ge2__power__minus__self,axiom,
    ! [K2: 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))),K2))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xt,K2)) ) ).

% mono_ge2_power_minus_self
tff(fact_9085_sum__list__upt,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_list_nat_nat(groups4561878855575611511st_nat,upt(M2,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N)) ) ) ).

% sum_list_upt
tff(fact_9086_map__Suc__upt,axiom,
    ! [M2: nat,N: nat] : map_nat_nat(suc,upt(M2,N)) = upt(aa_nat_nat(suc,M2),aa_nat_nat(suc,N)) ).

% map_Suc_upt
tff(fact_9087_map__add__upt,axiom,
    ! [N: nat,M2: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xu,N),upt(zero_zero_nat,M2)) = upt(N,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) ).

% map_add_upt
tff(fact_9088_map__decr__upt,axiom,
    ! [M2: nat,N: nat] : map_nat_nat(aTP_Lamm_xv,upt(aa_nat_nat(suc,M2),aa_nat_nat(suc,N))) = upt(M2,N) ).

% map_decr_upt
tff(fact_9089_card__length__sum__list__rec,axiom,
    ! [M2: nat,N5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M2))
     => ( finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_xw(M2),N5))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_xx(M2),N5)))),finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_xy(M2),N5)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9090_card__length__sum__list,axiom,
    ! [M2: nat,N5: nat] : finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_xw(M2),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),M2)),one_one_nat)),N5) ).

% card_length_sum_list
tff(fact_9091_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xz,N),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N) ) ) ).

% range_mod
tff(fact_9092_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_9093_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_9094_sorted__upt,axiom,
    ! [M2: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),upt(M2,N))) ).

% sorted_upt
tff(fact_9095_surj__prod__decode,axiom,
    image_5846123807819985514at_nat(nat_prod_decode,top_top_set_nat) = top_to4669805908274784177at_nat ).

% surj_prod_decode
tff(fact_9096_bij__prod__decode,axiom,
    bij_be8693218025023041337at_nat(nat_prod_decode,top_top_set_nat,top_to4669805908274784177at_nat) ).

% bij_prod_decode
tff(fact_9097_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_9098_bij__list__decode,axiom,
    bij_be6293887246118711976st_nat(nat_list_decode,top_top_set_nat,top_top_set_list_nat) ).

% bij_list_decode
tff(fact_9099_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_9100_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_9101_surj__prod__encode,axiom,
    image_2486076414777270412at_nat(nat_prod_encode,top_to4669805908274784177at_nat) = top_top_set_nat ).

% surj_prod_encode
tff(fact_9102_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_9103_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I2: 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,I2),aa_list_nat_nat(size_size_list_nat,Ns)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),aa_nat_nat(nth_nat(Ns),I2))) ) ) ).

% sorted_wrt_less_idx
tff(fact_9104_root__def,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = zero_zero_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_ya,N)),X) ) ) ) ).

% root_def
tff(fact_9105_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( 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(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),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,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9106_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( 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))),N))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
             => ( D3 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),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))),N))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
               => ( D3 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),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))),N))
               => ( D3 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),D3,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9107_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_yb,suminf_real(aa_real_fun_nat_real(aTP_Lamm_yc,X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9108_DERIV__fun__pow,axiom,
    ! [G2: fun_real_real,M2: real,X: real,N: nat] :
      ( has_fi5821293074295781190e_real(G2,M2,topolo2177554685111907308n_real(X,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_yd(G2),N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G2,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))),M2),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_9109_has__real__derivative__powr,axiom,
    ! [Z2: real,R: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z2))
     => has_fi5821293074295781190e_real(aTP_Lamm_ye(R),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(powr_real(Z2),aa_real_real(minus_minus_real(R),one_one_real))),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_9110_DERIV__series_H,axiom,
    ! [F2: fun_re4238438383274954274t_real,F4: fun_re4238438383274954274t_real,X0: real,A: real,B: real,L5: fun_nat_real] :
      ( ! [N3: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_yf(F2),N3),aa_nat_real(aa_real_fun_nat_real(F4,X0),N3),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X3: real] :
            ( pp(aa_set_real_bool(member_real(X3),set_or1633881224788618240n_real(A,B)))
           => summable_real(aa_real_fun_nat_real(F2,X3)) )
       => ( pp(aa_set_real_bool(member_real(X0),set_or1633881224788618240n_real(A,B)))
         => ( summable_real(aa_real_fun_nat_real(F4,X0))
           => ( summable_real(L5)
             => ( ! [N3: nat,X3: real,Y3: real] :
                    ( pp(aa_set_real_bool(member_real(X3),set_or1633881224788618240n_real(A,B)))
                   => ( pp(aa_set_real_bool(member_real(Y3),set_or1633881224788618240n_real(A,B)))
                     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(F2,X3),N3)),aa_nat_real(aa_real_fun_nat_real(F2,Y3),N3)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(L5,N3)),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X3),Y3))))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_yg(F2),suminf_real(aa_real_fun_nat_real(F4,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_9111_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(log2(B),aa_real_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_9112_DERIV__fun__powr,axiom,
    ! [G2: fun_real_real,M2: real,X: real,R: real] :
      ( has_fi5821293074295781190e_real(G2,M2,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G2,X)))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_yh(G2),R),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(powr_real(aa_real_real(G2,X)),aa_real_real(minus_minus_real(R),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M2),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9113_DERIV__powr,axiom,
    ! [G2: fun_real_real,M2: real,X: real,F2: fun_real_real,R: real] :
      ( has_fi5821293074295781190e_real(G2,M2,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G2,X)))
       => ( has_fi5821293074295781190e_real(F2,R,topolo2177554685111907308n_real(X,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_yi(G2),F2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(powr_real(aa_real_real(G2,X)),aa_real_real(F2,X))),aa_real_real(plus_plus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(ln_ln_real,aa_real_real(G2,X)))),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M2),aa_real_real(F2,X))),aa_real_real(G2,X)))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_9114_DERIV__power__series_H,axiom,
    ! [R2: real,F2: fun_nat_real,X0: real] :
      ( ! [X3: real] :
          ( pp(aa_set_real_bool(member_real(X3),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R2),R2)))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_yj(F2),X3)) )
     => ( pp(aa_set_real_bool(member_real(X0),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R2),R2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
         => has_fi5821293074295781190e_real(aTP_Lamm_yl(F2),suminf_real(aa_real_fun_nat_real(aTP_Lamm_yj(F2),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9115_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9116_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F2: fun_real_real,X: real,N: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
     => ( ! [M: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
       => ? [T5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F2,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ym(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9117_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F2: fun_real_real,X: real,N: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
        & ! [M: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),X3),topolo2177554685111907308n_real(X3,top_top_set_real)) )
     => ? [T5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,X)))
          & ( aa_real_real(F2,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ym(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9118_DERIV__odd__real__root,axiom,
    ! [N: 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))),N))
     => ( ( X != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9119_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,F2: 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),N))
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
         => ( ! [M: nat,T5: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),T5))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T5),zero_zero_real)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
           => ? [T5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),T5))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),zero_zero_real))
                & ( aa_real_real(F2,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_vf(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9120_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F2: fun_real_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
       => ( ! [M: nat,T5: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
                & 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(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ? [T5: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T5))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T5),H))
              & ( aa_real_real(F2,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_vf(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9121_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,F2: 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),N))
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
         => ( ! [M: nat,T5: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
                  & 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(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
           => ? [T5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T5))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),H))
                & ( aa_real_real(F2,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_vf(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9122_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F2: fun_real_real,N: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( X != zero_zero_real )
         => ( ! [M: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
           => ? [T5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T5)))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,X)))
                & ( aa_real_real(F2,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ym(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9123_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F2: fun_real_real,N: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
     => ( ! [M: nat,T5: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,X))) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
       => ? [T5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F2,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ym(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9124_Taylor__down,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F2: fun_real_real,A: real,B: real,C2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
       => ( ! [M: nat,T5: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T5))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T5),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),B))
             => ? [T5: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),T5))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),C2))
                  & ( aa_real_real(F2,A) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yn(Diff),A),C2)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(A),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_9125_Taylor__up,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F2: fun_real_real,A: real,B: real,C2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
       => ( ! [M: nat,T5: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T5))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T5),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B))
             => ? [T5: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),T5))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),B))
                  & ( aa_real_real(F2,B) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yn(Diff),B),C2)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(B),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9126_Taylor,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F2: fun_real_real,A: real,B: real,C2: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
       => ( ! [M: nat,T5: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T5))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T5),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),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 != C2 )
                   => ? [T5: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C2))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T5))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),C2)) ) )
                        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C2))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),T5))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T5),X)) ) )
                        & ( aa_real_real(F2,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yo(Diff),C2),X)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T5)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(X),C2)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9127_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun_na6724076748692969250l_real,K2: nat,B5: real] :
      ( ! [M: nat,T5: real] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
            & 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(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
     => ( ( N = aa_nat_nat(suc,K2) )
       => ! [M3: nat,T6: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),H)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_yq(N),Diff),B5),M3),aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6)),aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_yr(Diff),M3),T6)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M3))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B5),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,T6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M3)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M3))))))),topolo2177554685111907308n_real(T6,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9128_DERIV__pow,axiom,
    ! [N: nat,X: real,S2: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ys,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X,S2)) ).

% DERIV_pow
tff(fact_9129_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F2: fun_real_real,K2: real] :
      ( ( A != B )
     => ( ! [X3: real] : has_fi5821293074295781190e_real(F2,K2,topolo2177554685111907308n_real(X3,top_top_set_real))
       => ( aa_real_real(minus_minus_real(aa_real_real(F2,B)),aa_real_real(F2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),A)),K2) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_9130_MVT2,axiom,
    ! [A: real,B: real,F2: fun_real_real,F4: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
             => has_fi5821293074295781190e_real(F2,aa_real_real(F4,X3),topolo2177554685111907308n_real(X3,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(minus_minus_real(aa_real_real(F2,B)),aa_real_real(F2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),A)),aa_real_real(F4,Z)) ) ) ) ) ).

% MVT2
tff(fact_9131_GMVT_H,axiom,
    ! [A: real,B: real,F2: fun_real_real,G2: fun_real_real,G3: 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),F2) ) )
       => ( ! [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),G2) ) )
         => ( ! [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(G2,aa_real_real(G3,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(F2,aa_real_real(F4,Z),topolo2177554685111907308n_real(Z,top_top_set_real)) ) )
             => ? [C: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(aa_real_real(F2,B)),aa_real_real(F2,A))),aa_real_real(G3,C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(aa_real_real(G2,B)),aa_real_real(G2,A))),aa_real_real(F4,C)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9132_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))
         => ! [N7: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_yt(A))),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)),one_one_nat))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_9133_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)))
         => ! [N7: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_yt(A))),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)),one_one_nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_9134_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N3))),aa_nat_real(A,N3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(aTP_Lamm_yt(A))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yt(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))),N)),one_one_nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_9135_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_9136_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C2))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yu,C2),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9137_nested__sequence__unique,axiom,
    ! [F2: fun_nat_real,G2: fun_nat_real] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F2,N3)),aa_nat_real(F2,aa_nat_nat(suc,N3))))
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(G2,aa_nat_nat(suc,N3))),aa_nat_real(G2,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F2,N3)),aa_nat_real(G2,N3)))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_yv(F2),G2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L3: real] :
                ( ! [N7: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F2,N7)),L3))
                & filterlim_nat_real(F2,topolo2815343760600316023s_real(L3),at_top_nat)
                & ! [N7: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),aa_nat_real(G2,N7)))
                & filterlim_nat_real(G2,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_9138_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun_nat_real] :
      ( ! [R3: real] :
        ? [N6: nat] :
        ! [N3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N6),N3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R3),aa_nat_real(X6,N3))) )
     => filterlim_nat_real(aTP_Lamm_yw(X6),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9139_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_yx,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_9140_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_yy,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_9141_increasing__LIMSEQ,axiom,
    ! [F2: fun_nat_real,L: real] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F2,N3)),aa_nat_real(F2,aa_nat_nat(suc,N3))))
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F2,N3)),L))
       => ( ! [E2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
             => ? [N7: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),aa_real_real(plus_plus_real(aa_nat_real(F2,N7)),E2))) )
         => filterlim_nat_real(F2,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9142_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_yz,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_9143_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_za,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_9144_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_yt(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_9145_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N3))),aa_nat_real(A,N3)))
         => summable_real(aTP_Lamm_yt(A)) ) ) ) ).

% summable
tff(fact_9146_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zb(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_zc(Theta),K),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_9147_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_zd(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_zc(Theta),K),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_9148_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_ze(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_yt(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_9149_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(aa_real_fun_nat_real(aTP_Lamm_wa,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9150_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N3))),aa_nat_real(A,N3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yt(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))),N)))),suminf_real(aTP_Lamm_yt(A)))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9151_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N3))),aa_nat_real(A,N3)))
         => filterlim_nat_real(aTP_Lamm_ze(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_yt(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9152_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N3))),aa_nat_real(A,N3)))
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N3)))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L3: real] :
              ( ! [N7: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)))),L3))
              & filterlim_nat_real(aTP_Lamm_ze(A),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N7: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)),one_one_nat)))))
              & filterlim_nat_real(aTP_Lamm_zf(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9153_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_zf(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_yt(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_9154_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N3)))
       => ( ! [N3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N3))),aa_nat_real(A,N3)))
         => filterlim_nat_real(aTP_Lamm_zf(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_yt(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9155_real__bounded__linear,axiom,
    ! [F2: fun_real_real] :
      ( real_V5970128139526366754l_real(F2)
    <=> ? [C3: real] :
        ! [X4: real] : aa_real_real(F2,X4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X4),C3) ) ).

% real_bounded_linear
tff(fact_9156_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_9157_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim_real_real(aTP_Lamm_zg(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_9158_GMVT,axiom,
    ! [A: real,B: real,F2: fun_real_real,G2: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X3: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B)) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),F2) )
       => ( ! [X3: real] :
              ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B)) )
             => differ6690327859849518006l_real(F2,topolo2177554685111907308n_real(X3,top_top_set_real)) )
         => ( ! [X3: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B)) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),G2) )
           => ( ! [X3: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B)) )
                 => differ6690327859849518006l_real(G2,topolo2177554685111907308n_real(X3,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C: real] :
                  ( has_fi5821293074295781190e_real(G2,G_c,topolo2177554685111907308n_real(C,top_top_set_real))
                  & has_fi5821293074295781190e_real(F2,F_c,topolo2177554685111907308n_real(C,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_real,C),B))
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(aa_real_real(F2,B)),aa_real_real(F2,A))),G_c) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(aa_real_real(G2,B)),aa_real_real(G2,A))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_9159_eventually__sequentially__Suc,axiom,
    ! [P2: fun_nat_bool] :
      ( eventually_nat(aTP_Lamm_wm(P2),at_top_nat)
    <=> eventually_nat(P2,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_9160_eventually__sequentially,axiom,
    ! [P2: fun_nat_bool] :
      ( eventually_nat(P2,at_top_nat)
    <=> ? [N8: nat] :
        ! [N4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N4))
         => pp(aa_nat_bool(P2,N4)) ) ) ).

% eventually_sequentially
tff(fact_9161_eventually__sequentiallyI,axiom,
    ! [C2: nat,P2: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C2),X3))
         => pp(aa_nat_bool(P2,X3)) )
     => eventually_nat(P2,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_9162_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ord_le2510731241096832064er_nat(F3,at_top_nat)
    <=> ! [N8: nat] : eventually_nat(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),F3) ) ).

% le_sequentially
tff(fact_9163_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_9164_atLeast__Suc__greaterThan,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K2)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K2) ).

% atLeast_Suc_greaterThan
tff(fact_9165_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_9166_greaterThan__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_or1210151606488870762an_nat,K2)),insert_nat(aa_nat_nat(suc,K2),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_9167_atLeast__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_atLeast_nat,K2)),insert_nat(K2,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_9168_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F2: fun_real_real,F3: filter_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( filterlim_real_real(F2,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))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zh(N),F2),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_9169_GreatestI__ex__nat,axiom,
    ! [P2: fun_nat_bool,B: nat] :
      ( ? [X_1: nat] : pp(aa_nat_bool(P2,X_1))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P2,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B)) )
       => pp(aa_nat_bool(P2,order_Greatest_nat(P2))) ) ) ).

% GreatestI_ex_nat
tff(fact_9170_Greatest__le__nat,axiom,
    ! [P2: fun_nat_bool,K2: nat,B: nat] :
      ( pp(aa_nat_bool(P2,K2))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P2,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,K2),order_Greatest_nat(P2))) ) ) ).

% Greatest_le_nat
tff(fact_9171_GreatestI__nat,axiom,
    ! [P2: fun_nat_bool,K2: nat,B: nat] :
      ( pp(aa_nat_bool(P2,K2))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P2,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B)) )
       => pp(aa_nat_bool(P2,order_Greatest_nat(P2))) ) ) ).

% GreatestI_nat
tff(fact_9172_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F2: fun_real_real,F3: filter_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( filterlim_real_real(F2,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))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zh(N),F2),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9173_MVT,axiom,
    ! [A: real,B: real,F2: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F2)
       => ( ! [X3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B))
               => differ6690327859849518006l_real(F2,topolo2177554685111907308n_real(X3,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(F2,L3,topolo2177554685111907308n_real(Z,top_top_set_real))
              & ( aa_real_real(minus_minus_real(aa_real_real(F2,B)),aa_real_real(F2,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),A)),L3) ) ) ) ) ) ).

% MVT
tff(fact_9174_inj__sgn__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_real_bool(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_ya,N)),top_top_set_real)) ) ).

% inj_sgn_power
tff(fact_9175_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_9176_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_9177_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_9178_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9179_inj__list__encode,axiom,
    ! [A4: set_list_nat] : inj_on_list_nat_nat(nat_list_encode,A4) ).

% inj_list_encode
tff(fact_9180_inj__list__decode,axiom,
    ! [A4: set_nat] : pp(aa_set_nat_bool(inj_on_nat_list_nat(nat_list_decode),A4)) ).

% inj_list_decode
tff(fact_9181_inj__prod__encode,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : inj_on2178005380612969504at_nat(nat_prod_encode,A4) ).

% inj_prod_encode
tff(fact_9182_inj__Suc,axiom,
    ! [N5: set_nat] : pp(aa_set_nat_bool(inj_on_nat_nat(suc),N5)) ).

% inj_Suc
tff(fact_9183_inj__prod__decode,axiom,
    ! [A4: set_nat] : pp(aa_set_nat_bool(inj_on5538052773655684606at_nat(nat_prod_decode),A4)) ).

% inj_prod_decode
tff(fact_9184_inj__on__diff__nat,axiom,
    ! [N5: set_nat,K2: nat] :
      ( ! [N3: nat] :
          ( pp(aa_set_nat_bool(member_nat(N3),N5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N3)) )
     => pp(aa_set_nat_bool(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xe,K2)),N5)) ) ).

% inj_on_diff_nat
tff(fact_9185_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_9186_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_9187_Gcd__eq__Max,axiom,
    ! [M5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M5))
     => ( ( M5 != bot_bot_set_nat )
       => ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M5))
         => ( aa_set_nat_nat(gcd_Gcd_nat,M5) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_zi,M5))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9188_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_fr,N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_9189_Sup__nat__def,axiom,
    ! [X6: set_nat] :
      ( ( ( X6 = bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X6) = zero_zero_nat ) )
      & ( ( X6 != bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X6) = aa_set_nat_nat(lattic8265883725875713057ax_nat,X6) ) ) ) ).

% Sup_nat_def
tff(fact_9190_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_9191_divide__nat__def,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_zj(N),M2))) ) ) ) ).

% divide_nat_def
tff(fact_9192_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 )
                    & ! [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))))) )
                    & pp(aa_opt6800557945374683724t_bool(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_zk(Deg2),TreeList),Summary2))),Mima)) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9193_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 )
                  & ! [X5: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
                     => pp(aa_nat_bool(vEBT_VEBT_valid(X5),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))))) )
                  & pp(aa_opt6800557945374683724t_bool(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_zk(Deg2),TreeList),Summary2))),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9194_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 )
                & ! [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))))) )
                & pp(aa_opt6800557945374683724t_bool(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_zk(Deg2),TreeList),Summary2))),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9195_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList2,Summary)),Deg3))
    <=> ( ( Deg = Deg3 )
        & ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
           => pp(aa_nat_bool(vEBT_VEBT_valid(X4),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))))) )
        & pp(aa_opt6800557945374683724t_bool(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_zk(Deg),TreeList2),Summary))),Mima2)) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9196_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 )
                    & ! [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))))) )
                    & pp(aa_opt6800557945374683724t_bool(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_zk(Deg2),TreeList),Summary2))),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9197_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 )
                      & ! [X5: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X5),set_VEBT_VEBT2(TreeList)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X5),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))))) )
                      & pp(aa_opt6800557945374683724t_bool(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_zk(Deg2),TreeList),Summary2))),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9198_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 )
                      & ! [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))))) )
                      & pp(aa_opt6800557945374683724t_bool(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_zk(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_9199_min__Suc__Suc,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N)) ).

% min_Suc_Suc
tff(fact_9200_min__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N) = zero_zero_nat ).

% min_0L
tff(fact_9201_min__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9202_take__bit__num__simps_I1_J,axiom,
    ! [M2: num] : bit_take_bit_num(zero_zero_nat,M2) = none_num ).

% take_bit_num_simps(1)
tff(fact_9203_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] : bit_take_bit_num(aa_nat_nat(suc,N),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_9204_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M2: num] : bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit0,M2)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_zl,bit_take_bit_num(N,M2)) ).

% take_bit_num_simps(3)
tff(fact_9205_min__numeral__Suc,axiom,
    ! [K2: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K2)),N)) ).

% min_numeral_Suc
tff(fact_9206_min__Suc__numeral,axiom,
    ! [N: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),aa_num_nat(pred_numeral,K2))) ).

% min_Suc_numeral
tff(fact_9207_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M2: num] : bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit1,M2)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(N,M2))) ).

% take_bit_num_simps(4)
tff(fact_9208_min__diff,axiom,
    ! [M2: nat,I2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),I2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N)),I2) ).

% min_diff
tff(fact_9209_nat__mult__min__left,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).

% nat_mult_min_left
tff(fact_9210_nat__mult__min__right,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Q2)) ).

% nat_mult_min_right
tff(fact_9211_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9212_min__Suc1,axiom,
    ! [N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),M2) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_zm,N)),M2) ).

% min_Suc1
tff(fact_9213_min__Suc2,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_zn,N)),M2) ).

% min_Suc2
tff(fact_9214_take__bit__num__def,axiom,
    ! [N: nat,M2: num] :
      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M2)) = zero_zero_nat )
       => ( bit_take_bit_num(N,M2) = none_num ) )
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M2)) != zero_zero_nat )
       => ( bit_take_bit_num(N,M2) = some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M2)))) ) ) ) ).

% take_bit_num_def
tff(fact_9215_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_zo)) ).

% pred_nat_def
tff(fact_9216_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_zp) ).

% Rats_eq_int_div_nat
tff(fact_9217_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_9218_divmod__integer__eq__cases,axiom,
    ! [K2: code_integer,L: code_integer] :
      ( ( ( K2 = zero_z3403309356797280102nteger )
       => ( code_divmod_integer(K2,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
      & ( ( K2 != zero_z3403309356797280102nteger )
       => ( ( ( L = zero_z3403309356797280102nteger )
           => ( code_divmod_integer(K2,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K2) ) )
          & ( ( L != zero_z3403309356797280102nteger )
           => ( code_divmod_integer(K2,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,K2)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L)),code_divmod_abs(K2,L),produc6916734918728496179nteger(aTP_Lamm_zq(L),code_divmod_abs(K2,L)))) ) ) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_9219_less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M2),N),transi6264000038957366511cl_nat(pred_nat))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% less_eq
tff(fact_9220_pred__nat__trancl__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M2),N),transi2905341329935302413cl_nat(pred_nat))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% pred_nat_trancl_eq_le
tff(fact_9221_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_9222_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_9223_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_9224_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_9225_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_9226_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_9227_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_9228_gcd__Suc__0,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_9229_gcd__pos__nat,axiom,
    ! [M2: nat,N: 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,M2),N)))
    <=> ( ( M2 != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9230_gcd__diff2__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),N) ) ) ).

% gcd_diff2_nat
tff(fact_9231_gcd__diff1__nat,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),N) ) ) ).

% gcd_diff1_nat
tff(fact_9232_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_9233_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_9234_gcd__mult__distrib__nat,axiom,
    ! [K2: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N)) ).

% gcd_mult_distrib_nat
tff(fact_9235_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_9236_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_9237_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_9238_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X3: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).

% bezout_nat
tff(fact_9239_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X3: 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),X3)))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),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),X3)))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9240_Field__natLeq__on,axiom,
    ! [N: nat] : field_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_zr(N)))) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hm,N)) ).

% Field_natLeq_on
tff(fact_9241_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_zs) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9242_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),N) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_zt(N),M2))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9243_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_9244_gcd__mult__distrib__int,axiom,
    ! [K2: int,M2: int,N: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,K2)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M2),N)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),M2)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),N)) ).

% gcd_mult_distrib_int
tff(fact_9245_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U2: int,V: 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,V),Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).

% bezout_int
tff(fact_9246_strict__mono__imp__increasing,axiom,
    ! [F2: fun_nat_nat,N: nat] :
      ( order_5726023648592871131at_nat(F2)
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(F2,N))) ) ).

% strict_mono_imp_increasing
tff(fact_9247_DeMoivre2,axiom,
    ! [R: real,A: real,N: nat] : aa_nat_complex(power_power_complex(aa_real_complex(rcis(R),A)),N) = aa_real_complex(rcis(aa_nat_real(aa_real_fun_nat_real(power_power_real,R),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),A)) ).

% DeMoivre2
tff(fact_9248_Re__rcis,axiom,
    ! [R: real,A: real] : aa_complex_real(re,aa_real_complex(rcis(R),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(cos_real,A)) ).

% Re_rcis
tff(fact_9249_Im__rcis,axiom,
    ! [R: real,A: real] : aa_complex_real(im,aa_real_complex(rcis(R),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(sin_real,A)) ).

% Im_rcis
tff(fact_9250_rcis__mult,axiom,
    ! [R1: real,A: real,R22: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(rcis(R1),A)),aa_real_complex(rcis(R22),B)) = aa_real_complex(rcis(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R1),R22)),aa_real_real(plus_plus_real(A),B)) ).

% rcis_mult
tff(fact_9251_rcis__def,axiom,
    ! [R: real,A: real] : aa_real_complex(rcis(R),A) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R)),aa_real_complex(cis,A)) ).

% rcis_def
tff(fact_9252_eventually__prod__sequentially,axiom,
    ! [P2: fun_Pr7757163693017731481t_bool] :
      ( eventu1038000079068216329at_nat(P2,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N8: nat] :
        ! [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M6))
         => ! [N4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N4))
             => pp(aa_Pro2232559814647311250t_bool(P2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N4),M6))) ) ) ) ).

% eventually_prod_sequentially
tff(fact_9253_sqr_Osimps_I3_J,axiom,
    ! [N: num] : aa_num_num(sqr,aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(sqr,N)),N))) ).

% sqr.simps(3)
tff(fact_9254_sqr_Osimps_I2_J,axiom,
    ! [N: num] : aa_num_num(sqr,aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(sqr,N))) ).

% sqr.simps(2)
tff(fact_9255_sqr_Osimps_I1_J,axiom,
    aa_num_num(sqr,one) = one ).

% sqr.simps(1)
tff(fact_9256_sqr__conv__mult,axiom,
    ! [X: num] : aa_num_num(sqr,X) = aa_num_num(aa_num_fun_num_num(times_times_num,X),X) ).

% sqr_conv_mult
tff(fact_9257_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,X),aa_num_num(bit1,Y)) = aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(sqr,aa_num_num(aa_num_fun_num_num(pow,X),Y))),X) ).

% pow.simps(3)
tff(fact_9258_pow_Osimps_I1_J,axiom,
    ! [X: num] : aa_num_num(aa_num_fun_num_num(pow,X),one) = X ).

% pow.simps(1)
tff(fact_9259_pow_Osimps_I2_J,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,X),aa_num_num(bit0,Y)) = aa_num_num(sqr,aa_num_num(aa_num_fun_num_num(pow,X),Y)) ).

% pow.simps(2)
tff(fact_9260_natLeq__on__wo__rel,axiom,
    ! [N: nat] : bNF_We3818239936649020644el_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_zr(N)))) ).

% natLeq_on_wo_rel
tff(fact_9261_pairs__le__eq__Sigma,axiom,
    ! [M2: nat] : collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_zu(M2))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,M2),aTP_Lamm_zv(M2)) ).

% pairs_le_eq_Sigma
tff(fact_9262_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_9263_coprime__Suc__0__right,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(suc,zero_zero_nat))) ).

% coprime_Suc_0_right
tff(fact_9264_coprime__Suc__0__left,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ).

% coprime_Suc_0_left
tff(fact_9265_coprime__Suc__left__nat,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(suc,N)),N)) ).

% coprime_Suc_left_nat
tff(fact_9266_coprime__Suc__right__nat,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(suc,N))) ).

% coprime_Suc_right_nat
tff(fact_9267_coprime__crossproduct__nat,axiom,
    ! [A: nat,D: nat,B: nat,C2: 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),C2))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D) )
        <=> ( ( A = B )
            & ( C2 = D ) ) ) ) ) ).

% coprime_crossproduct_nat
tff(fact_9268_vimage__Suc__insert__Suc,axiom,
    ! [N: nat,A4: set_nat] : vimage_nat_nat(suc,insert_nat(aa_nat_nat(suc,N),A4)) = insert_nat(N,vimage_nat_nat(suc,A4)) ).

% vimage_Suc_insert_Suc
tff(fact_9269_vimage__Suc__insert__0,axiom,
    ! [A4: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A4)) = vimage_nat_nat(suc,A4) ).

% vimage_Suc_insert_0
tff(fact_9270_coprime__diff__one__left__nat,axiom,
    ! [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(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N)) ) ).

% coprime_diff_one_left_nat
tff(fact_9271_coprime__diff__one__right__nat,axiom,
    ! [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(algebr934650988132801477me_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_9272_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( pp(aa_set_real_bool(member_real(X),field_5140801741446780682s_real))
     => ~ ! [M: nat,N3: nat] :
            ( ( N3 != zero_zero_nat )
           => ( ( aa_real_real(abs_abs_real,X) = aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N3)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,M),N3)) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_9273_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_9274_set__encode__vimage__Suc,axiom,
    ! [A4: set_nat] : aa_set_nat_nat(nat_set_encode,vimage_nat_nat(suc,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(nat_set_encode,A4)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% set_encode_vimage_Suc
tff(fact_9275_Restr__natLeq,axiom,
    ! [N: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hm,N)),aTP_Lamm_zw(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_zr(N))) ).

% Restr_natLeq
tff(fact_9276_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_9277_coprime__crossproduct__int,axiom,
    ! [A: int,D: int,B: int,C2: 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),C2))
       => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,C2)) = 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,C2) = aa_int_int(abs_abs_int,D) ) ) ) ) ) ).

% coprime_crossproduct_int
tff(fact_9278_natLeq__Linear__order,axiom,
    order_4473980167227706203on_nat(field_nat(bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Linear_order
tff(fact_9279_natLeq__Total,axiom,
    total_on_nat(field_nat(bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Total
tff(fact_9280_natLeq__Refl,axiom,
    refl_on_nat(field_nat(bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Refl
tff(fact_9281_natLeq__natLess__Id,axiom,
    bNF_Ca8459412986667044542atLess = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,bNF_Ca8665028551170535155natLeq),id_nat2) ).

% natLeq_natLess_Id
tff(fact_9282_Field__natLeq,axiom,
    field_nat(bNF_Ca8665028551170535155natLeq) = top_top_set_nat ).

% Field_natLeq
tff(fact_9283_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_9284_Restr__natLeq2,axiom,
    ! [N: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N),aTP_Lamm_zx(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_zr(N))) ).

% Restr_natLeq2
tff(fact_9285_natLeq__underS__less,axiom,
    ! [N: nat] : aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hm,N)) ).

% natLeq_underS_less
tff(fact_9286_pair__lessI2,axiom,
    ! [A: nat,B: nat,S2: 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,S2),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_9287_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_9288_pair__lessI1,axiom,
    ! [A: nat,B: nat,S2: 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),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_9289_pair__leqI2,axiom,
    ! [A: nat,B: nat,S2: 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,S2),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_9290_pair__leqI1,axiom,
    ! [A: nat,B: nat,S2: 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),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_9291_length__upto,axiom,
    ! [I2: int,J2: int] : size_size_list_int(upto(I2,J2)) = 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,J2),I2)),one_one_int)) ).

% length_upto
tff(fact_9292_gcd__nat_Oordering__top__axioms,axiom,
    pp(aa_nat_bool(ordering_top_nat(dvd_dvd_nat,aTP_Lamm_zs),zero_zero_nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_9293_bot__nat__0_Oordering__top__axioms,axiom,
    pp(aa_nat_bool(ordering_top_nat(aTP_Lamm_ah,aTP_Lamm_hm),zero_zero_nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_9294_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_9295_ratrel__def,axiom,
    ! [X5: product_prod_int_int,Xa: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X5),Xa))
    <=> ( ( product_snd_int_int(X5) != 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(X5)),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(X5)) ) ) ) ).

% ratrel_def
tff(fact_9296_plus__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod_int_int,X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,Xa2),Xa2))
     => ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa2)),aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xa2)),product_snd_int_int(X))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X)),product_snd_int_int(Xa2))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Xa2)),product_snd_int_int(X)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_9297_times__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod_int_int,X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,Xa2),Xa2))
     => ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa2)),aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xa2)),product_fst_int_int(X)),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Xa2)),product_snd_int_int(X)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_9298_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_9299_Rat_Opositive__mult,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(positive,X))
     => ( pp(aa_rat_bool(positive,Y))
       => pp(aa_rat_bool(positive,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y))) ) ) ).

% Rat.positive_mult
tff(fact_9300_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_9301_Rat_Opositive_Orsp,axiom,
    bNF_re1394333895916556125l_bool(ratrel,fequal_bool,aTP_Lamm_zy,aTP_Lamm_zy) ).

% Rat.positive.rsp
tff(fact_9302_Rat_Opositive__def,axiom,
    positive = map_fu5118516228767806217l_bool(rep_Rat,id_bool,aTP_Lamm_zy) ).

% Rat.positive_def
tff(fact_9303_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_9304_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_9305_times__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),times_times_int,times_times_int) ).

% times_integer.rsp
tff(fact_9306_times__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),times_times_nat,times_times_nat) ).

% times_natural.rsp
tff(fact_9307_times__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_zz,aTP_Lamm_zz) ).

% times_rat.rsp
tff(fact_9308_plus__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_aaa,aTP_Lamm_aaa) ).

% plus_rat.rsp
tff(fact_9309_plus__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_aaa,plus_plus_rat) ).

% plus_rat.transfer
tff(fact_9310_Rat_Opositive_Otransfer,axiom,
    bNF_re3535474257078034868l_bool(pcr_rat,fequal_bool,aTP_Lamm_zy,positive) ).

% Rat.positive.transfer
tff(fact_9311_times__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_zz,times_times_rat) ).

% times_rat.transfer
tff(fact_9312_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_ww),times_times_int) ).

% times_int.transfer
tff(fact_9313_minus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_xd),minus_minus_int) ).

% minus_int.transfer
tff(fact_9314_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_9315_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_aab,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_9316_uminus__int_Otransfer,axiom,
    pp(aa_fun_int_int_bool(aa_fun7527181275634263258t_bool(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_wx)),uminus_uminus_int)) ).

% uminus_int.transfer
tff(fact_9317_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_9318_less__eq__int_Otransfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_wz),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_9319_plus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_xb),plus_plus_int) ).

% plus_int.transfer
tff(fact_9320_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_ww),produc4528391901023682385at_nat(aTP_Lamm_ww)) ).

% times_int.rsp
tff(fact_9321_minus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_xd),produc4528391901023682385at_nat(aTP_Lamm_xd)) ).

% minus_int.rsp
tff(fact_9322_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V2: 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),V2)))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),V2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,U),Y) ) ) ).

% intrel_iff
tff(fact_9323_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_9324_uminus__int_Orsp,axiom,
    pp(aa_fun4519828053337421206t_bool(aa_fun2399787666232550060t_bool(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_wx)),produc2626176000494625587at_nat(aTP_Lamm_wx))) ).

% uminus_int.rsp
tff(fact_9325_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_9326_less__eq__int_Orsp,axiom,
    bNF_re2219331435504962769t_bool(intrel,bNF_re3189209160384574813l_bool(intrel,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_wz),produc5023667602222594023t_bool(aTP_Lamm_wz)) ).

% less_eq_int.rsp
tff(fact_9327_plus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_xb),produc4528391901023682385at_nat(aTP_Lamm_xb)) ).

% plus_int.rsp
tff(fact_9328_less__eq__enat__def,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( ord_le2932123472753598470d_enat(M2,N)
    <=> extend8126323780198025762t_bool(aTP_Lamm_aac(M2),fTrue,N) ) ).

% less_eq_enat_def
tff(fact_9329_natLeq__Well__order,axiom,
    order_2888998067076097458on_nat(field_nat(bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Well_order
tff(fact_9330_natLeq__on__well__order__on,axiom,
    ! [N: nat] : order_2888998067076097458on_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hm,N)),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_zr(N)))) ).

% natLeq_on_well_order_on
tff(fact_9331_natLeq__on__Well__order,axiom,
    ! [N: nat] : order_2888998067076097458on_nat(field_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_zr(N)))),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_zr(N)))) ).

% natLeq_on_Well_order
tff(fact_9332_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_9333_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_9334_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_9335_plus__rat__def,axiom,
    plus_plus_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_aaa) ).

% plus_rat_def
tff(fact_9336_diff__rat,axiom,
    ! [B: int,D: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C2),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% diff_rat
tff(fact_9337_sgn__rat,axiom,
    ! [A: int,B: int] : aa_rat_rat(sgn_sgn_rat,aa_int_rat(fract(A),B)) = aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B))) ).

% sgn_rat
tff(fact_9338_mult__rat,axiom,
    ! [A: int,B: int,C2: int,D: int] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C2),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ).

% mult_rat
tff(fact_9339_divide__rat,axiom,
    ! [A: int,B: int,C2: int,D: int] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C2),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% divide_rat
tff(fact_9340_less__rat,axiom,
    ! [B: int,D: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C2),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,C2),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)))) ) ) ) ).

% less_rat
tff(fact_9341_add__rat,axiom,
    ! [B: int,D: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C2),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% add_rat
tff(fact_9342_le__rat,axiom,
    ! [B: int,D: int,A: int,C2: 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(fract(A),B)),aa_int_rat(fract(C2),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,C2),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)))) ) ) ) ).

% le_rat
tff(fact_9343_mult__rat__cancel,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) = aa_int_rat(fract(A),B) ) ) ).

% mult_rat_cancel
tff(fact_9344_eq__rat_I1_J,axiom,
    ! [B: int,D: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( aa_int_rat(fract(A),B) = aa_int_rat(fract(C2),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,C2),B) ) ) ) ) ).

% eq_rat(1)
tff(fact_9345_positive__rat,axiom,
    ! [A: int,B: int] :
      ( pp(aa_rat_bool(positive,aa_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_9346_times__rat__def,axiom,
    times_times_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_zz) ).

% times_rat_def
tff(fact_9347_integer__of__char__code,axiom,
    ! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool] : integer_of_char(char2(B0,B1,B22,B32,B42,B52,B62,B72)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(B72)),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B62))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B52))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B42))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B32))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B22))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B1))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B0)) ).

% integer_of_char_code
tff(fact_9348_char_Osize_I2_J,axiom,
    ! [X1: bool,X2: bool,X33: bool,X42: bool,X52: bool,X62: bool,X7: bool,X8: bool] : size_size_char(char2(X1,X2,X33,X42,X52,X62,X7,X8)) = zero_zero_nat ).

% char.size(2)
tff(fact_9349_char_Osize__gen,axiom,
    ! [X1: bool,X2: bool,X33: bool,X42: bool,X52: bool,X62: bool,X7: bool,X8: bool] : size_char(char2(X1,X2,X33,X42,X52,X62,X7,X8)) = zero_zero_nat ).

% char.size_gen
tff(fact_9350_numeral__le__enat__iff,axiom,
    ! [M2: num,N: nat] :
      ( ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M2),extended_enat2(N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M2)),N)) ) ).

% numeral_le_enat_iff
tff(fact_9351_idiff__enat__0__right,axiom,
    ! [N: extended_enat] : minus_3235023915231533773d_enat(N,extended_enat2(zero_zero_nat)) = N ).

% idiff_enat_0_right
tff(fact_9352_idiff__enat__0,axiom,
    ! [N: extended_enat] : minus_3235023915231533773d_enat(extended_enat2(zero_zero_nat),N) = extended_enat2(zero_zero_nat) ).

% idiff_enat_0
tff(fact_9353_times__enat__simps_I1_J,axiom,
    ! [M2: nat,N: nat] : times_7803423173614009249d_enat(extended_enat2(M2),extended_enat2(N)) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) ).

% times_enat_simps(1)
tff(fact_9354_enat__ord__simps_I1_J,axiom,
    ! [M2: nat,N: nat] :
      ( ord_le2932123472753598470d_enat(extended_enat2(M2),extended_enat2(N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% enat_ord_simps(1)
tff(fact_9355_Suc__ile__eq,axiom,
    ! [M2: nat,N: extended_enat] :
      ( ord_le2932123472753598470d_enat(extended_enat2(aa_nat_nat(suc,M2)),N)
    <=> ord_le72135733267957522d_enat(extended_enat2(M2),N) ) ).

% Suc_ile_eq
tff(fact_9356_enat__0__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( zero_z5237406670263579293d_enat = extended_enat2(X) )
    <=> ( X = zero_zero_nat ) ) ).

% enat_0_iff(2)
tff(fact_9357_enat__0__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( extended_enat2(X) = zero_z5237406670263579293d_enat )
    <=> ( X = zero_zero_nat ) ) ).

% enat_0_iff(1)
tff(fact_9358_zero__enat__def,axiom,
    zero_z5237406670263579293d_enat = extended_enat2(zero_zero_nat) ).

% zero_enat_def
tff(fact_9359_iadd__le__enat__iff,axiom,
    ! [X: extended_enat,Y: extended_enat,N: nat] :
      ( ord_le2932123472753598470d_enat(plus_p3455044024723400733d_enat(X,Y),extended_enat2(N))
    <=> ? [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)),N)) ) ) ).

% iadd_le_enat_iff
tff(fact_9360_elimnum,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary)),N))
     => ( 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))),N))) = vEBT_Node(Info2,Deg,TreeList2,Summary) ) ) ).

% elimnum
tff(fact_9361_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_aad(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_9362_enat__0__less__mult__iff,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,times_7803423173614009249d_enat(M2,N))
    <=> ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,M2)
        & ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,N) ) ) ).

% enat_0_less_mult_iff
tff(fact_9363_imult__is__0,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( ( times_7803423173614009249d_enat(M2,N) = zero_z5237406670263579293d_enat )
    <=> ( ( M2 = zero_z5237406670263579293d_enat )
        | ( N = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_9364_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_9365_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_aad(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_aad(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_9366_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_aad(Deg),TreeList2),vEBT_VEBT_elim_dead(Summary,extend5688581933313929465d_enat)) ).

% VEBT_internal.elim_dead.simps(2)
tff(fact_9367_elimcomplete,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary)),N))
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info2,Deg,TreeList2,Summary),extend5688581933313929465d_enat) = vEBT_Node(Info2,Deg,TreeList2,Summary) ) ) ).

% elimcomplete
tff(fact_9368_times__enat__simps_I2_J,axiom,
    times_7803423173614009249d_enat(extend5688581933313929465d_enat,extend5688581933313929465d_enat) = extend5688581933313929465d_enat ).

% times_enat_simps(2)
tff(fact_9369_times__enat__simps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( times_7803423173614009249d_enat(extend5688581933313929465d_enat,extended_enat2(N)) = zero_z5237406670263579293d_enat ) )
      & ( ( N != zero_zero_nat )
       => ( times_7803423173614009249d_enat(extend5688581933313929465d_enat,extended_enat2(N)) = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(3)
tff(fact_9370_times__enat__simps_I4_J,axiom,
    ! [M2: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( times_7803423173614009249d_enat(extended_enat2(M2),extend5688581933313929465d_enat) = zero_z5237406670263579293d_enat ) )
      & ( ( M2 != zero_zero_nat )
       => ( times_7803423173614009249d_enat(extended_enat2(M2),extend5688581933313929465d_enat) = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(4)
tff(fact_9371_imult__is__infinity,axiom,
    ! [A: extended_enat,B: extended_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_9372_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_9373_imult__infinity,axiom,
    ! [N: extended_enat] :
      ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,N)
     => ( times_7803423173614009249d_enat(extend5688581933313929465d_enat,N) = extend5688581933313929465d_enat ) ) ).

% imult_infinity
tff(fact_9374_imult__infinity__right,axiom,
    ! [N: extended_enat] :
      ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,N)
     => ( times_7803423173614009249d_enat(N,extend5688581933313929465d_enat) = extend5688581933313929465d_enat ) ) ).

% imult_infinity_right
tff(fact_9375_times__enat__def,axiom,
    ! [M2: extended_enat,N: extended_enat] : times_7803423173614009249d_enat(M2,N) = extend3600170679010898289d_enat(aTP_Lamm_aaf(N),if_Extended_enat(fequal_Extended_enat(N,zero_z5237406670263579293d_enat),zero_z5237406670263579293d_enat,extend5688581933313929465d_enat),M2) ).

% times_enat_def
tff(fact_9376_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_aad(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_aad(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_9377_eSuc__def,axiom,
    ! [I2: extended_enat] : extended_eSuc(I2) = extend3600170679010898289d_enat(aTP_Lamm_aag,extend5688581933313929465d_enat,I2) ).

% eSuc_def
tff(fact_9378_binomial__def,axiom,
    ! [N: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K2) = finite_card_set_nat(collect_set_nat(aa_nat8857836526276733660t_bool(aTP_Lamm_aah(N),K2))) ).

% binomial_def
tff(fact_9379_enat__eSuc__iff,axiom,
    ! [Y: nat,X: extended_enat] :
      ( ( extended_enat2(Y) = extended_eSuc(X) )
    <=> ? [N4: nat] :
          ( ( Y = aa_nat_nat(suc,N4) )
          & ( extended_enat2(N4) = X ) ) ) ).

% enat_eSuc_iff
tff(fact_9380_eSuc__enat__iff,axiom,
    ! [X: extended_enat,Y: nat] :
      ( ( extended_eSuc(X) = extended_enat2(Y) )
    <=> ? [N4: nat] :
          ( ( Y = aa_nat_nat(suc,N4) )
          & ( X = extended_enat2(N4) ) ) ) ).

% eSuc_enat_iff
tff(fact_9381_eSuc__enat,axiom,
    ! [N: nat] : extended_eSuc(extended_enat2(N)) = extended_enat2(aa_nat_nat(suc,N)) ).

% eSuc_enat
tff(fact_9382_mult__eSuc__right,axiom,
    ! [M2: extended_enat,N: extended_enat] : times_7803423173614009249d_enat(M2,extended_eSuc(N)) = plus_p3455044024723400733d_enat(M2,times_7803423173614009249d_enat(M2,N)) ).

% mult_eSuc_right
tff(fact_9383_mult__eSuc,axiom,
    ! [M2: extended_enat,N: extended_enat] : times_7803423173614009249d_enat(extended_eSuc(M2),N) = plus_p3455044024723400733d_enat(N,times_7803423173614009249d_enat(M2,N)) ).

% mult_eSuc
tff(fact_9384_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( 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_9385_Real_Opositive_Orsp,axiom,
    bNF_re2717234250851183977l_bool(realrel,fequal_bool,aTP_Lamm_aai,aTP_Lamm_aai) ).

% Real.positive.rsp
tff(fact_9386_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_9387_times__real_Orsp,axiom,
    bNF_re769765379126235449at_rat(realrel,bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_aaj,aTP_Lamm_aaj) ).

% times_real.rsp
tff(fact_9388_Real_Opositive_Otransfer,axiom,
    bNF_re1458715823057739248l_bool(pcr_real,fequal_bool,aTP_Lamm_aai,positive2) ).

% Real.positive.transfer
tff(fact_9389_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_9390_times__real_Otransfer,axiom,
    bNF_re1162444993429354318l_real(pcr_real,bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_aaj,times_times_real) ).

% times_real.transfer
tff(fact_9391_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_9392_Real_Opositive__def,axiom,
    positive2 = map_fu1722713901878115497l_bool(rep_real,id_bool,aTP_Lamm_aai) ).

% Real.positive_def
tff(fact_9393_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,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_9394_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,real2(Xa2)),real2(X)) = real2(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_aaj,Xa2),X)) ) ) ) ).

% times_real.abs_eq
tff(fact_9395_le__Real,axiom,
    ! [X6: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,real2(X6)),real2(Y6)))
        <=> ! [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(X6,N4)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Y6,N4)),R5))) ) ) ) ) ) ).

% le_Real
tff(fact_9396_not__positive__Real,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( ~ pp(aa_real_bool(positive2,real2(X6)))
      <=> ! [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(X6,N4)),R5)) ) ) ) ) ).

% not_positive_Real
tff(fact_9397_cauchy__mult,axiom,
    ! [X6: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => cauchy(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_aaj,X6),Y6)) ) ) ).

% cauchy_mult
tff(fact_9398_mult__Real,axiom,
    ! [X6: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,real2(X6)),real2(Y6)) = real2(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_aaj,X6),Y6)) ) ) ) ).

% mult_Real
tff(fact_9399_cauchyD,axiom,
    ! [X6: fun_nat_rat,R: rat] :
      ( cauchy(X6)
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R))
       => ? [K: nat] :
          ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M3))
           => ! [N7: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N7))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(X6,M3)),aa_nat_rat(X6,N7)))),R)) ) ) ) ) ).

% cauchyD
tff(fact_9400_cauchyI,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [R3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R3))
         => ? [K4: nat] :
            ! [M: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),M))
             => ! [N3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N3))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(X6,M)),aa_nat_rat(X6,N3)))),R3)) ) ) )
     => cauchy(X6) ) ).

% cauchyI
tff(fact_9401_cauchy__def,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
    <=> ! [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(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(X6,M6)),aa_nat_rat(X6,N4)))),R5)) ) ) ) ) ).

% cauchy_def
tff(fact_9402_positive__Real,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( pp(aa_real_bool(positive2,real2(X6)))
      <=> ? [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(X6,N4))) ) ) ) ) ).

% positive_Real
tff(fact_9403_cauchy__not__vanishes__cases,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ? [B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
            & ? [K: nat] :
                ( ! [N7: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N7))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(X6,N7)))) )
                | ! [N7: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N7))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_nat_rat(X6,N7))) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_9404_cauchy__not__vanishes,axiom,
    ! [X6: fun_nat_rat] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ? [B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
            & ? [K: nat] :
              ! [N7: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N7))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N7)))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_9405_vanishesD,axiom,
    ! [X6: fun_nat_rat,R: rat] :
      ( vanishes(X6)
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R))
       => ? [K: nat] :
          ! [N7: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N7))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N7))),R)) ) ) ) ).

% vanishesD
tff(fact_9406_vanishesI,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [R3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R3))
         => ? [K4: nat] :
            ! [N3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N3))),R3)) ) )
     => vanishes(X6) ) ).

% vanishesI
tff(fact_9407_vanishes__def,axiom,
    ! [X6: fun_nat_rat] :
      ( vanishes(X6)
    <=> ! [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(X6,N4))),R5)) ) ) ) ).

% vanishes_def
tff(fact_9408_vanishes__mult__bounded,axiom,
    ! [X6: fun_nat_rat,Y6: fun_nat_rat] :
      ( ? [A6: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A6))
          & ! [N3: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X6,N3))),A6)) )
     => ( vanishes(Y6)
       => vanishes(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_aaj,X6),Y6)) ) ) ).

% vanishes_mult_bounded
tff(fact_9409_surj__int__decode,axiom,
    image_nat_int(nat_int_decode,top_top_set_nat) = top_top_set_int ).

% surj_int_decode
tff(fact_9410_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_9411_inj__int__decode,axiom,
    ! [A4: set_nat] : pp(aa_set_nat_bool(inj_on_nat_int(nat_int_decode),A4)) ).

% inj_int_decode
tff(fact_9412_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_9413_nat__to__rat__surj__def,axiom,
    ! [N: nat] : aa_nat_rat(nat_to_rat_surj,N) = produc6207742614233964070at_rat(aTP_Lamm_aak,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ).

% nat_to_rat_surj_def
tff(fact_9414_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_9415_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_9416_int__decode__inverse,axiom,
    ! [N: nat] : aa_int_nat(nat_int_encode,aa_nat_int(nat_int_decode,N)) = N ).

% int_decode_inverse
tff(fact_9417_inj__int__encode,axiom,
    ! [A4: set_int] : pp(aa_set_int_bool(inj_on_int_nat(nat_int_encode),A4)) ).

% inj_int_encode
tff(fact_9418_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_9419_surj__int__encode,axiom,
    image_int_nat(nat_int_encode,top_top_set_int) = top_top_set_nat ).

% surj_int_encode
tff(fact_9420_linear__scale__real,axiom,
    ! [F2: fun_real_real,R: real,B: real] :
      ( real_V4572627801940501177l_real(F2)
     => ( aa_real_real(F2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(F2,B)) ) ) ).

% linear_scale_real
tff(fact_9421_real__linearD,axiom,
    ! [F2: fun_real_real] :
      ( real_V4572627801940501177l_real(F2)
     => ~ ! [C: real] : F2 != aa_rea7340605660300375871l_real(times_times_real,C) ) ).

% real_linearD
tff(fact_9422_natLeq__Partial__order,axiom,
    order_5251275573222108571on_nat(field_nat(bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Partial_order
tff(fact_9423_natLeq__Preorder,axiom,
    order_4861654808422542329on_nat(field_nat(bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Preorder
tff(fact_9424_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_wz)) ).

% less_eq_int_def
tff(fact_9425_int__encode__def,axiom,
    ! [I2: int] : aa_int_nat(nat_int_encode,I2) = aa_Sum8073061460641146765at_nat(nat_sum_encode,if_Sum_sum_nat_nat(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I2),sum_Inl_nat_nat(aa_int_nat(nat2,I2)),sum_Inr_nat_nat(aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,I2)),one_one_int))))) ).

% int_encode_def
tff(fact_9426_surj__sum__encode,axiom,
    image_1320371278474632150at_nat(nat_sum_encode,top_to6661820994512907621at_nat) = top_top_set_nat ).

% surj_sum_encode
tff(fact_9427_inj__sum__encode,axiom,
    ! [A4: set_Sum_sum_nat_nat] : inj_on6343450744447823682at_nat(nat_sum_encode,A4) ).

% inj_sum_encode
tff(fact_9428_sum__encode__eq,axiom,
    ! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] :
      ( ( aa_Sum8073061460641146765at_nat(nat_sum_encode,X) = aa_Sum8073061460641146765at_nat(nat_sum_encode,Y) )
    <=> ( X = Y ) ) ).

% sum_encode_eq
tff(fact_9429_bij__sum__encode,axiom,
    pp(aa_set_nat_bool(bij_be5432664580149595207at_nat(nat_sum_encode,top_to6661820994512907621at_nat),top_top_set_nat)) ).

% bij_sum_encode
tff(fact_9430_le__sum__encode__Inl,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,X),aa_Sum8073061460641146765at_nat(nat_sum_encode,sum_Inl_nat_nat(Y)))) ) ).

% le_sum_encode_Inl
tff(fact_9431_le__sum__encode__Inr,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,X),aa_Sum8073061460641146765at_nat(nat_sum_encode,sum_Inr_nat_nat(Y)))) ) ).

% le_sum_encode_Inr
tff(fact_9432_sum__decode__def,axiom,
    ! [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))
       => ( aa_nat7431386967378518541at_nat(nat_sum_decode,N) = sum_Inl_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),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))),N))
       => ( aa_nat7431386967378518541at_nat(nat_sum_decode,N) = sum_Inr_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% sum_decode_def
tff(fact_9433_sum__encode__inverse,axiom,
    ! [X: sum_sum_nat_nat] : aa_nat7431386967378518541at_nat(nat_sum_decode,aa_Sum8073061460641146765at_nat(nat_sum_encode,X)) = X ).

% sum_encode_inverse
tff(fact_9434_sum__decode__inverse,axiom,
    ! [N: nat] : aa_Sum8073061460641146765at_nat(nat_sum_encode,aa_nat7431386967378518541at_nat(nat_sum_decode,N)) = N ).

% sum_decode_inverse
tff(fact_9435_bij__sum__decode,axiom,
    bij_be4790990086886966983at_nat(nat_sum_decode,top_top_set_nat,top_to6661820994512907621at_nat) ).

% bij_sum_decode
tff(fact_9436_sum__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,X) = aa_nat7431386967378518541at_nat(nat_sum_decode,Y) )
    <=> ( X = Y ) ) ).

% sum_decode_eq
tff(fact_9437_inj__sum__decode,axiom,
    ! [A4: set_nat] : pp(aa_set_nat_bool(inj_on5701776251185195458at_nat(nat_sum_decode),A4)) ).

% inj_sum_decode
tff(fact_9438_surj__sum__decode,axiom,
    image_678696785212003926at_nat(nat_sum_decode,top_top_set_nat) = top_to6661820994512907621at_nat ).

% surj_sum_decode
tff(fact_9439_nth__item_Opinduct,axiom,
    ! [A0: nat,P2: fun_nat_bool] :
      ( pp(aa_nat_bool(accp_nat(nth_item_rel),A0))
     => ( ( pp(aa_nat_bool(accp_nat(nth_item_rel),zero_zero_nat))
         => pp(aa_nat_bool(P2,zero_zero_nat)) )
       => ( ! [N3: nat] :
              ( pp(aa_nat_bool(accp_nat(nth_item_rel),aa_nat_nat(suc,N3)))
             => ( ! [A6: nat,Aa: nat] :
                    ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,N3) = sum_Inl_nat_nat(A6) )
                   => ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,A6) = sum_Inl_nat_nat(Aa) )
                     => pp(aa_nat_bool(P2,Aa)) ) )
               => ( ! [A6: nat,B6: nat] :
                      ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,N3) = sum_Inl_nat_nat(A6) )
                     => ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,A6) = sum_Inr_nat_nat(B6) )
                       => pp(aa_nat_bool(P2,B6)) ) )
                 => ( ! [B6: nat,Ba: nat,X5: nat,Y4: nat] :
                        ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,N3) = sum_Inr_nat_nat(B6) )
                       => ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,B6) = sum_Inr_nat_nat(Ba) )
                         => ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y4) = aa_nat7603221163087113075at_nat(nat_prod_decode,Ba) )
                           => pp(aa_nat_bool(P2,X5)) ) ) )
                   => ( ! [B6: nat,Ba: nat,X5: nat,Y4: nat] :
                          ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,N3) = sum_Inr_nat_nat(B6) )
                         => ( ( aa_nat7431386967378518541at_nat(nat_sum_decode,B6) = sum_Inr_nat_nat(Ba) )
                           => ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y4) = aa_nat7603221163087113075at_nat(nat_prod_decode,Ba) )
                             => pp(aa_nat_bool(P2,Y4)) ) ) )
                     => pp(aa_nat_bool(P2,aa_nat_nat(suc,N3))) ) ) ) ) )
         => pp(aa_nat_bool(P2,A0)) ) ) ) ).

% nth_item.pinduct
tff(fact_9440_int__decode__def,axiom,
    ! [N: nat] : aa_nat_int(nat_int_decode,N) = sum_ca7763040182479039464nt_nat(semiri1314217659103216013at_int,aTP_Lamm_aal,aa_nat7431386967378518541at_nat(nat_sum_decode,N)) ).

% int_decode_def
tff(fact_9441_sum__encode__def,axiom,
    ! [X: sum_sum_nat_nat] : aa_Sum8073061460641146765at_nat(nat_sum_encode,X) = sum_ca6763686470577984908at_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aTP_Lamm_aam,X) ).

% sum_encode_def
tff(fact_9442_enat__def,axiom,
    ! [N: nat] : extended_enat2(N) = extended_Abs_enat(some_nat(N)) ).

% enat_def
tff(fact_9443_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F2: fun_nat_real,M5: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_gp(F2),M5),at_top_nat)
     => ( ! [M: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),M))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F2,N3)),aa_nat_real(F2,M))) ) )
       => topolo7531315842566124627t_real(F2) ) ) ).

% Bseq_monoseq_convergent'_dec
tff(fact_9444_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F2: fun_nat_real,M5: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_gp(F2),M5),at_top_nat)
     => ( ! [M: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),M))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F2,M)),aa_nat_real(F2,N3))) ) )
       => topolo7531315842566124627t_real(F2) ) ) ).

% Bseq_monoseq_convergent'_inc
tff(fact_9445_Bseq__mono__convergent,axiom,
    ! [X6: fun_nat_real] :
      ( bfun_nat_real(X6,at_top_nat)
     => ( ! [M: nat,N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X6,M)),aa_nat_real(X6,N3))) )
       => topolo7531315842566124627t_real(X6) ) ) ).

% Bseq_mono_convergent
tff(fact_9446_lcm__altdef__int,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(gcd_lcm_int,A),B) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)) ).

% lcm_altdef_int
tff(fact_9447_prod__gcd__lcm__int,axiom,
    ! [M2: int,N: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,M2)),aa_int_int(abs_abs_int,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M2),N)),aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M2),N)) ).

% prod_gcd_lcm_int
tff(fact_9448_lcm__0__iff__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M2),N) = zero_zero_nat )
    <=> ( ( M2 = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% lcm_0_iff_nat
tff(fact_9449_lcm__1__iff__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M2),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M2 = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% lcm_1_iff_nat
tff(fact_9450_prod__gcd__lcm__nat,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),N)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M2),N)) ).

% prod_gcd_lcm_nat
tff(fact_9451_lcm__pos__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( 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(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M2),N))) ) ) ).

% lcm_pos_nat
tff(fact_9452_lcm__code__integer,axiom,
    ! [A: code_integer,B: code_integer] : gcd_lcm_Code_integer(A,B) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)),gcd_gcd_Code_integer(A,B)) ).

% lcm_code_integer
tff(fact_9453_lcm__nat__def,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) ).

% lcm_nat_def
tff(fact_9454_Lcm__eq__Max__nat,axiom,
    ! [M5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M5))
     => ( ( M5 != bot_bot_set_nat )
       => ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M5))
         => ( ! [M: nat,N3: nat] :
                ( pp(aa_set_nat_bool(member_nat(M),M5))
               => ( pp(aa_set_nat_bool(member_nat(N3),M5))
                 => pp(aa_set_nat_bool(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M),N3)),M5)) ) )
           => ( aa_set_nat_nat(gcd_Lcm_nat,M5) = aa_set_nat_nat(lattic8265883725875713057ax_nat,M5) ) ) ) ) ) ).

% Lcm_eq_Max_nat
tff(fact_9455_Lcm__eq__0__I__nat,axiom,
    ! [A4: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4))
     => ( aa_set_nat_nat(gcd_Lcm_nat,A4) = zero_zero_nat ) ) ).

% Lcm_eq_0_I_nat
tff(fact_9456_Lcm__0__iff__nat,axiom,
    ! [A4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A4))
     => ( ( aa_set_nat_nat(gcd_Lcm_nat,A4) = zero_zero_nat )
      <=> pp(aa_set_nat_bool(member_nat(zero_zero_nat),A4)) ) ) ).

% Lcm_0_iff_nat
tff(fact_9457_Lcm__nat__infinite,axiom,
    ! [M5: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,M5))
     => ( aa_set_nat_nat(gcd_Lcm_nat,M5) = zero_zero_nat ) ) ).

% Lcm_nat_infinite
tff(fact_9458_Lcm__nat__def,axiom,
    ! [M5: set_nat] :
      ( ( pp(aa_set_nat_bool(finite_finite_nat,M5))
       => ( aa_set_nat_nat(gcd_Lcm_nat,M5) = aa_set_nat_nat(lattic7826324295020591184_F_nat(gcd_lcm_nat,one_one_nat),M5) ) )
      & ( ~ pp(aa_set_nat_bool(finite_finite_nat,M5))
       => ( aa_set_nat_nat(gcd_Lcm_nat,M5) = zero_zero_nat ) ) ) ).

% Lcm_nat_def
tff(fact_9459_unit__factor__simps_I1_J,axiom,
    aa_nat_nat(unit_f2748546683901255202or_nat,zero_zero_nat) = zero_zero_nat ).

% unit_factor_simps(1)
tff(fact_9460_unit__factor__simps_I2_J,axiom,
    ! [N: nat] : aa_nat_nat(unit_f2748546683901255202or_nat,aa_nat_nat(suc,N)) = one_one_nat ).

% unit_factor_simps(2)
tff(fact_9461_unit__factor__nat__def,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(unit_f2748546683901255202or_nat,N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(unit_f2748546683901255202or_nat,N) = one_one_nat ) ) ) ).

% unit_factor_nat_def
tff(fact_9462_times__num__def,axiom,
    ! [M2: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,M2),N) = aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(nat_of_num,M2)),aa_num_nat(nat_of_num,N))) ).

% times_num_def
tff(fact_9463_nat__of__num__code_I2_J,axiom,
    ! [N: num] : aa_num_nat(nat_of_num,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,N)),aa_num_nat(nat_of_num,N)) ).

% nat_of_num_code(2)
tff(fact_9464_less__eq__num__def,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M2),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(nat_of_num,M2)),aa_num_nat(nat_of_num,N))) ) ).

% less_eq_num_def
tff(fact_9465_less__num__def,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(nat_of_num,M2)),aa_num_nat(nat_of_num,N))) ) ).

% less_num_def
tff(fact_9466_nat__of__num__inc,axiom,
    ! [X: num] : aa_num_nat(nat_of_num,aa_num_num(inc,X)) = aa_nat_nat(suc,aa_num_nat(nat_of_num,X)) ).

% nat_of_num_inc
tff(fact_9467_nat__of__num__mult,axiom,
    ! [X: num,Y: num] : aa_num_nat(nat_of_num,aa_num_num(aa_num_fun_num_num(times_times_num,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(nat_of_num,X)),aa_num_nat(nat_of_num,Y)) ).

% nat_of_num_mult
tff(fact_9468_nat__of__num__numeral,axiom,
    nat_of_num = numeral_numeral_nat ).

% nat_of_num_numeral
tff(fact_9469_num__eq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
    <=> ( aa_num_nat(nat_of_num,X) = aa_num_nat(nat_of_num,Y) ) ) ).

% num_eq_iff
tff(fact_9470_nat__of__num__inverse,axiom,
    ! [X: num] : aa_nat_num(num_of_nat,aa_num_nat(nat_of_num,X)) = X ).

% nat_of_num_inverse
tff(fact_9471_nat__of__num_Osimps_I2_J,axiom,
    ! [X: num] : aa_num_nat(nat_of_num,aa_num_num(bit0,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,X)),aa_num_nat(nat_of_num,X)) ).

% nat_of_num.simps(2)
tff(fact_9472_nat__of__num__pos,axiom,
    ! [X: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(nat_of_num,X))) ).

% nat_of_num_pos
tff(fact_9473_nat__of__num__neq__0,axiom,
    ! [X: num] : aa_num_nat(nat_of_num,X) != zero_zero_nat ).

% nat_of_num_neq_0
tff(fact_9474_nat__of__num__code_I1_J,axiom,
    aa_num_nat(nat_of_num,one) = one_one_nat ).

% nat_of_num_code(1)
tff(fact_9475_nat__of__num__add,axiom,
    ! [X: num,Y: num] : aa_num_nat(nat_of_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,X)),aa_num_nat(nat_of_num,Y)) ).

% nat_of_num_add
tff(fact_9476_nat__of__num__sqr,axiom,
    ! [X: num] : aa_num_nat(nat_of_num,aa_num_num(sqr,X)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(nat_of_num,X)),aa_num_nat(nat_of_num,X)) ).

% nat_of_num_sqr
tff(fact_9477_nat__of__num_Osimps_I1_J,axiom,
    aa_num_nat(nat_of_num,one) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_of_num.simps(1)
tff(fact_9478_nat__of__num_Osimps_I3_J,axiom,
    ! [X: num] : aa_num_nat(nat_of_num,aa_num_num(bit1,X)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,X)),aa_num_nat(nat_of_num,X))) ).

% nat_of_num.simps(3)
tff(fact_9479_num__of__nat__inverse,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_num_nat(nat_of_num,aa_nat_num(num_of_nat,N)) = N ) ) ).

% num_of_nat_inverse
tff(fact_9480_nat__of__num__code_I3_J,axiom,
    ! [N: num] : aa_num_nat(nat_of_num,aa_num_num(bit1,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,N)),aa_num_nat(nat_of_num,N))) ).

% nat_of_num_code(3)
tff(fact_9481_plus__num__def,axiom,
    ! [M2: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,M2),N) = aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(nat_of_num,M2)),aa_num_nat(nat_of_num,N))) ).

% plus_num_def
tff(fact_9482_natLeq__trans,axiom,
    trans_nat(bNF_Ca8665028551170535155natLeq) ).

% natLeq_trans
tff(fact_9483_natLeq__antisym,axiom,
    antisym_nat(bNF_Ca8665028551170535155natLeq) ).

% natLeq_antisym
tff(fact_9484_VEBT__internal_Olesseq_Oelims_I3_J,axiom,
    ! [X: option_nat,Xa2: option_nat] :
      ( ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,X),Xa2))
     => ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_eq_nat),X),Xa2)) ) ).

% VEBT_internal.lesseq.elims(3)
tff(fact_9485_VEBT__internal_Olesseq_Oelims_I2_J,axiom,
    ! [X: option_nat,Xa2: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,X),Xa2))
     => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_eq_nat),X),Xa2)) ) ).

% VEBT_internal.lesseq.elims(2)
tff(fact_9486_VEBT__internal_Olesseq_Oelims_I1_J,axiom,
    ! [X: option_nat,Xa2: option_nat,Y: bool] :
      ( ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,X),Xa2))
      <=> pp(Y) )
     => ( pp(Y)
      <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_eq_nat),X),Xa2)) ) ) ).

% VEBT_internal.lesseq.elims(1)
tff(fact_9487_VEBT__internal_Olesseq_Osimps,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,X),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_V2881884560877996034ft_nat(ord_less_eq_nat),X),Y)) ) ).

% VEBT_internal.lesseq.simps
tff(fact_9488_integer__of__nat__0,axiom,
    aa_nat_Code_integer(code_integer_of_nat,zero_zero_nat) = zero_z3403309356797280102nteger ).

% integer_of_nat_0
tff(fact_9489_natural__decr,axiom,
    ! [N: code_natural] :
      ( ( N != zero_z2226904508553997617atural )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Code_natural_nat(code_nat_of_natural,N)),aa_nat_nat(suc,zero_zero_nat))),aa_Code_natural_nat(code_nat_of_natural,N))) ) ).

% natural_decr
tff(fact_9490_times__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa2: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,X),Xa2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa2)) ).

% times_natural.rep_eq
tff(fact_9491_zero__natural_Orep__eq,axiom,
    aa_Code_natural_nat(code_nat_of_natural,zero_z2226904508553997617atural) = zero_zero_nat ).

% zero_natural.rep_eq
tff(fact_9492_less__eq__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa2: code_natural] :
      ( ord_le1926595141338095240atural(X,Xa2)
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa2))) ) ).

% less_eq_natural.rep_eq
tff(fact_9493_next_Osimps,axiom,
    ! [V2: code_natural,W2: code_natural] : aa_Pro7165575656943818355atural(next,produc3574140220909816553atural(V2,W2)) = aa_Pro7165575656943818355atural(produc6639722614265839536atural(plus_p4538020629002901425atural(minus_shift(numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,modulo8411746178871703098atural(V2,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(V2,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))))))))))))),plus_p4538020629002901425atural(minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,modulo8411746178871703098atural(W2,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(W2,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))),one_one_Code_natural)),one_one_Code_natural)),produc3574140220909816553atural(minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,modulo8411746178871703098atural(V2,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(V2,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))))))))))))),minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,modulo8411746178871703098atural(W2,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(W2,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))))) ).

% next.simps
tff(fact_9494_Random_Orange__def,axiom,
    ! [K2: code_natural] : range(K2) = produc5538323210962509403atural(iterat8892046348760725948atural(log(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),K2),aTP_Lamm_aao,one_one_Code_natural),aTP_Lamm_aap(K2)) ).

% Random.range_def
tff(fact_9495_Suc_Orep__eq,axiom,
    ! [X: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(code_Suc,X)) = aa_nat_nat(suc,aa_Code_natural_nat(code_nat_of_natural,X)) ).

% Suc.rep_eq
tff(fact_9496_Suc_Oabs__eq,axiom,
    ! [X: nat] : aa_Cod4272041906116450446atural(code_Suc,aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(suc,X)) ).

% Suc.abs_eq
tff(fact_9497_less__eq__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] :
      ( ord_le1926595141338095240atural(aa_nat_Code_natural(code_natural_of_nat,Xa2),aa_nat_Code_natural(code_natural_of_nat,X))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X)) ) ).

% less_eq_natural.abs_eq
tff(fact_9498_times__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_nat_Code_natural(code_natural_of_nat,Xa2)),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa2),X)) ).

% times_natural.abs_eq
tff(fact_9499_less__nat__rel,axiom,
    ord_less_nat = transi2163837189807498211lp_nat(aTP_Lamm_zo) ).

% less_nat_rel
tff(fact_9500_zero__natural__def,axiom,
    zero_z2226904508553997617atural = aa_nat_Code_natural(code_natural_of_nat,zero_zero_nat) ).

% zero_natural_def
tff(fact_9501_Code__Numeral_OSuc__def,axiom,
    code_Suc = aa_fun5114738747585492520atural(map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),suc) ).

% Code_Numeral.Suc_def
tff(fact_9502_times__natural__def,axiom,
    times_2397367101498566445atural = map_fu7613509917203888380atural(code_nat_of_natural,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),times_times_nat) ).

% times_natural_def
tff(fact_9503_typerep_Osize__neq,axiom,
    ! [X: typerep] : aa_typerep_nat(size_size_typerep,X) != zero_zero_nat ).

% typerep.size_neq
tff(fact_9504_typerep_Osize_I2_J,axiom,
    ! [X1: literal,X2: list_typerep] : aa_typerep_nat(size_size_typerep,typerep2(X1,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_typerep(size_size_typerep,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% typerep.size(2)
tff(fact_9505_typerep_Osize__gen,axiom,
    ! [X1: literal,X2: list_typerep] : aa_typerep_nat(size_typerep,typerep2(X1,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_typerep(size_typerep,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% typerep.size_gen
tff(fact_9506_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_9507_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_9508_real__times__code,axiom,
    ! [X: rat,Y: rat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_rat_real(ratreal,X)),aa_rat_real(ratreal,Y)) = aa_rat_real(ratreal,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)) ).

% real_times_code
tff(fact_9509_max__nat_Omonoid__axioms,axiom,
    pp(aa_nat_bool(monoid_nat(ord_max_nat),zero_zero_nat)) ).

% max_nat.monoid_axioms
tff(fact_9510_gcd__nat_Omonoid__axioms,axiom,
    pp(aa_nat_bool(monoid_nat(gcd_gcd_nat),zero_zero_nat)) ).

% gcd_nat.monoid_axioms
tff(fact_9511_natLeq__Card__order,axiom,
    bNF_Ca1281551314933786834on_nat(field_nat(bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Card_order
tff(fact_9512_card__of__nat,axiom,
    member8757157785044589968at_nat(produc2922128104949294807at_nat(bNF_Ca3793111618940312692of_nat(top_top_set_nat),bNF_Ca8665028551170535155natLeq),bNF_We5258908940166488438at_nat) ).

% card_of_nat
tff(fact_9513_card__of__Field__natLeq,axiom,
    member8757157785044589968at_nat(produc2922128104949294807at_nat(bNF_Ca3793111618940312692of_nat(field_nat(bNF_Ca8665028551170535155natLeq)),bNF_Ca8665028551170535155natLeq),bNF_We5258908940166488438at_nat) ).

% card_of_Field_natLeq
tff(fact_9514_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_wc,Uu) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_9515_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_hv,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_2
tff(fact_9516_ATP_Olambda__3,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_wk,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_3
tff(fact_9517_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_aal,Uu) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Uu))),one_one_int) ).

% ATP.lambda_4
tff(fact_9518_ATP_Olambda__5,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_vl,Uu) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu)) ).

% ATP.lambda_5
tff(fact_9519_ATP_Olambda__6,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_xv,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_6
tff(fact_9520_ATP_Olambda__7,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_kw,Uu) = plus_p5714425477246183910nteger(Uu,one_one_Code_integer) ).

% ATP.lambda_7
tff(fact_9521_ATP_Olambda__8,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_kx,Uu) = aa_complex_complex(plus_plus_complex(Uu),one_one_complex) ).

% ATP.lambda_8
tff(fact_9522_ATP_Olambda__9,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_kz,Uu) = aa_real_real(plus_plus_real(Uu),one_one_real) ).

% ATP.lambda_9
tff(fact_9523_ATP_Olambda__10,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_ky,Uu) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),one_one_rat) ).

% ATP.lambda_10
tff(fact_9524_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_lb,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_11
tff(fact_9525_ATP_Olambda__12,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_la,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_12
tff(fact_9526_ATP_Olambda__13,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_aab,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),zero_zero_nat) ).

% ATP.lambda_13
tff(fact_9527_ATP_Olambda__14,axiom,
    ! [Uu: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_zy,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_14
tff(fact_9528_ATP_Olambda__15,axiom,
    ! [Uu: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_aao,Uu) = produc5538323210962509403atural(next,aTP_Lamm_aan(Uu)) ).

% ATP.lambda_15
tff(fact_9529_ATP_Olambda__16,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_aam,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))),Uu)) ).

% ATP.lambda_16
tff(fact_9530_ATP_Olambda__17,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_yb,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_wb,Uu)) ).

% ATP.lambda_17
tff(fact_9531_ATP_Olambda__18,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_zi,Uu) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_fr,Uu)) ).

% ATP.lambda_18
tff(fact_9532_ATP_Olambda__19,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_yx,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_19
tff(fact_9533_ATP_Olambda__20,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_zl,Uu) = some_num(aa_num_num(bit0,Uu)) ).

% ATP.lambda_20
tff(fact_9534_ATP_Olambda__21,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_wh,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_21
tff(fact_9535_ATP_Olambda__22,axiom,
    ! [Uu: nat] : aa_nat_Extended_enat(aTP_Lamm_aag,Uu) = extended_enat2(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_22
tff(fact_9536_ATP_Olambda__23,axiom,
    ! [Uu: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aTP_Lamm_aai,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_23
tff(fact_9537_ATP_Olambda__24,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_zp,Uu))
    <=> ? [I: int,N4: nat] :
          ( ( Uu = aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,I)),aa_nat_real(semiri5074537144036343181t_real,N4)) )
          & ( N4 != zero_zero_nat ) ) ) ).

% ATP.lambda_24
tff(fact_9538_ATP_Olambda__25,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_op(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_25
tff(fact_9539_ATP_Olambda__26,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_or,Uu),Uua) = aa_real_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_26
tff(fact_9540_ATP_Olambda__27,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_oo(Uu),Uua) = aa_rat_rat(aa_rat_fun_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_27
tff(fact_9541_ATP_Olambda__28,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_oq,Uu),Uua) = aa_int_int(aa_int_fun_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_28
tff(fact_9542_ATP_Olambda__29,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_wu,Uu),Uua) = aa_int_int(aa_int_fun_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))),aa_Code_integer_int(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))),aa_Code_integer_int(code_int_of_integer,Uu))),one_one_int)) ).

% ATP.lambda_29
tff(fact_9543_ATP_Olambda__30,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_wt,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,aa_Code_integer_nat(code_nat_of_integer,Uu)),aa_Code_integer_nat(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,aa_Code_integer_nat(code_nat_of_integer,Uu)),aa_Code_integer_nat(code_nat_of_integer,Uu))),one_one_nat)) ).

% ATP.lambda_30
tff(fact_9544_ATP_Olambda__31,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_xf,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_31
tff(fact_9545_ATP_Olambda__32,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_aaf(Uu),Uua) = extend3600170679010898289d_enat(aTP_Lamm_aae(Uua),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_32
tff(fact_9546_ATP_Olambda__33,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_aac(Uu),Uua))
    <=> extend8126323780198025762t_bool(aa_nat_fun_nat_bool(aTP_Lamm_ah,Uua),fFalse,Uu) ) ).

% ATP.lambda_33
tff(fact_9547_ATP_Olambda__34,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ol(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_34
tff(fact_9548_ATP_Olambda__35,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_on,Uu),Uua) = aa_real_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_35
tff(fact_9549_ATP_Olambda__36,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_ok(Uu),Uua) = aa_rat_rat(aa_rat_fun_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_36
tff(fact_9550_ATP_Olambda__37,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_om,Uu),Uua) = aa_int_int(aa_int_fun_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_37
tff(fact_9551_ATP_Olambda__38,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_aaa,Uu),Uua) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu)),product_snd_int_int(Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uua)),product_snd_int_int(Uu))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Uu)),product_snd_int_int(Uua))) ).

% ATP.lambda_38
tff(fact_9552_ATP_Olambda__39,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_wd,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_39
tff(fact_9553_ATP_Olambda__40,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vj,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_40
tff(fact_9554_ATP_Olambda__41,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vz,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_41
tff(fact_9555_ATP_Olambda__42,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_uy(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(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_42
tff(fact_9556_ATP_Olambda__43,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ux(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_43
tff(fact_9557_ATP_Olambda__44,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_vb,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_44
tff(fact_9558_ATP_Olambda__45,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_uz(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(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_45
tff(fact_9559_ATP_Olambda__46,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_va,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_46
tff(fact_9560_ATP_Olambda__47,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_zz,Uu),Uua) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu)),product_fst_int_int(Uua)),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Uu)),product_snd_int_int(Uua))) ).

% ATP.lambda_47
tff(fact_9561_ATP_Olambda__48,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_wb,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_48
tff(fact_9562_ATP_Olambda__49,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yc,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_49
tff(fact_9563_ATP_Olambda__50,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_yt(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_50
tff(fact_9564_ATP_Olambda__51,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_tt(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_51
tff(fact_9565_ATP_Olambda__52,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ts(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_52
tff(fact_9566_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_tw,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_53
tff(fact_9567_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_tu(Uu),Uua) = aa_rat_rat(aa_rat_fun_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_54
tff(fact_9568_ATP_Olambda__55,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_tv,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_55
tff(fact_9569_ATP_Olambda__56,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_oz(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_56
tff(fact_9570_ATP_Olambda__57,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pb,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu),Uua)),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_57
tff(fact_9571_ATP_Olambda__58,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_pa(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_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_58
tff(fact_9572_ATP_Olambda__59,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_uh(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_59
tff(fact_9573_ATP_Olambda__60,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uj,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)) ).

% ATP.lambda_60
tff(fact_9574_ATP_Olambda__61,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ui(Uu),Uua) = aa_rat_rat(aa_rat_fun_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_61
tff(fact_9575_ATP_Olambda__62,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_zs,Uu),Uua))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uu),Uua))
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_62
tff(fact_9576_ATP_Olambda__63,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rv,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% ATP.lambda_63
tff(fact_9577_ATP_Olambda__64,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_we,Uu),Uua) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_64
tff(fact_9578_ATP_Olambda__65,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(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uua)),Uua) ).

% ATP.lambda_65
tff(fact_9579_ATP_Olambda__66,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qv,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_66
tff(fact_9580_ATP_Olambda__67,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_bl(Uu),Uua))
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_67
tff(fact_9581_ATP_Olambda__68,axiom,
    ! [Uu: nat,Uua: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_bk,Uu),Uua))
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_68
tff(fact_9582_ATP_Olambda__69,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_zg(Uu),Uua) = aa_real_real(powr_real(aa_real_real(plus_plus_real(one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),Uua))),aa_real_real(divide_divide_real(one_one_real),Uua)) ).

% ATP.lambda_69
tff(fact_9583_ATP_Olambda__70,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_wa,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_70
tff(fact_9584_ATP_Olambda__71,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_np(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_71
tff(fact_9585_ATP_Olambda__72,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_nq(Uu),Uua) = aa_rat_rat(aa_rat_fun_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_72
tff(fact_9586_ATP_Olambda__73,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ns(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_73
tff(fact_9587_ATP_Olambda__74,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_nr(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_74
tff(fact_9588_ATP_Olambda__75,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hl(Uu),Uua) = aa_real_real(plus_plus_real(aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_75
tff(fact_9589_ATP_Olambda__76,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_hi(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_76
tff(fact_9590_ATP_Olambda__77,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_hk(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_77
tff(fact_9591_ATP_Olambda__78,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hj(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_78
tff(fact_9592_ATP_Olambda__79,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ha(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_79
tff(fact_9593_ATP_Olambda__80,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_gy(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_80
tff(fact_9594_ATP_Olambda__81,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_gz(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,aa_nat_nat(suc,Uua))),aa_nat_int(Uu,Uua)) ).

% ATP.lambda_81
tff(fact_9595_ATP_Olambda__82,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ei(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_82
tff(fact_9596_ATP_Olambda__83,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ej(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).

% ATP.lambda_83
tff(fact_9597_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ek(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_84
tff(fact_9598_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hh(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_85
tff(fact_9599_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_hf(Uu),Uua) = aa_rat_rat(aa_rat_fun_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_86
tff(fact_9600_ATP_Olambda__87,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hg(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_87
tff(fact_9601_ATP_Olambda__88,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_hb(Uu),Uua) = aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_88
tff(fact_9602_ATP_Olambda__89,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_he(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_89
tff(fact_9603_ATP_Olambda__90,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_hc(Uu),Uua) = aa_rat_rat(aa_rat_fun_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_90
tff(fact_9604_ATP_Olambda__91,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hd(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_91
tff(fact_9605_ATP_Olambda__92,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lx(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_92
tff(fact_9606_ATP_Olambda__93,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lv(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_93
tff(fact_9607_ATP_Olambda__94,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lw(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_94
tff(fact_9608_ATP_Olambda__95,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_qt(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_95
tff(fact_9609_ATP_Olambda__96,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_qq(Uu),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_96
tff(fact_9610_ATP_Olambda__97,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_qn(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_97
tff(fact_9611_ATP_Olambda__98,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_qk(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_98
tff(fact_9612_ATP_Olambda__99,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_ge(Uu),Uua))
    <=> ( aa_complex_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_99
tff(fact_9613_ATP_Olambda__100,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_gh(Uu),Uua))
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_100
tff(fact_9614_ATP_Olambda__101,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_gk(Uu),Uua))
    <=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_101
tff(fact_9615_ATP_Olambda__102,axiom,
    ! [Uu: fun_real_complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_gc(Uu),Uua))
    <=> ( aa_real_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_102
tff(fact_9616_ATP_Olambda__103,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_gf(Uu),Uua))
    <=> ( aa_real_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_103
tff(fact_9617_ATP_Olambda__104,axiom,
    ! [Uu: fun_real_rat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_gi(Uu),Uua))
    <=> ( aa_real_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_104
tff(fact_9618_ATP_Olambda__105,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_gl(Uu),Uua))
    <=> ( aa_real_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_105
tff(fact_9619_ATP_Olambda__106,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_gd(Uu),Uua))
    <=> ( aa_int_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_106
tff(fact_9620_ATP_Olambda__107,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_gg(Uu),Uua))
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_107
tff(fact_9621_ATP_Olambda__108,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_gj(Uu),Uua))
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_108
tff(fact_9622_ATP_Olambda__109,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_zf(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yt(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_109
tff(fact_9623_ATP_Olambda__110,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ze(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_yt(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_110
tff(fact_9624_ATP_Olambda__111,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wi,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uu) ).

% ATP.lambda_111
tff(fact_9625_ATP_Olambda__112,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_dc(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_112
tff(fact_9626_ATP_Olambda__113,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_db,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_113
tff(fact_9627_ATP_Olambda__114,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ya,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_114
tff(fact_9628_ATP_Olambda__115,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vk,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_115
tff(fact_9629_ATP_Olambda__116,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vm,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_116
tff(fact_9630_ATP_Olambda__117,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_aak,Uu),Uua) = aa_int_rat(fract(aa_nat_int(nat_int_decode,Uu)),aa_nat_int(nat_int_decode,Uua)) ).

% ATP.lambda_117
tff(fact_9631_ATP_Olambda__118,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_sm(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu) ).

% ATP.lambda_118
tff(fact_9632_ATP_Olambda__119,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_so,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).

% ATP.lambda_119
tff(fact_9633_ATP_Olambda__120,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_sn(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_120
tff(fact_9634_ATP_Olambda__121,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_kh(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_121
tff(fact_9635_ATP_Olambda__122,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kj,Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Uua)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% ATP.lambda_122
tff(fact_9636_ATP_Olambda__123,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ki(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri681578069525770553at_rat,Uua)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% ATP.lambda_123
tff(fact_9637_ATP_Olambda__124,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sx,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_124
tff(fact_9638_ATP_Olambda__125,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_za,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_real_real(plus_plus_real(one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_125
tff(fact_9639_ATP_Olambda__126,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(aTP_Lamm_aad(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_126
tff(fact_9640_ATP_Olambda__127,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_wj,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_127
tff(fact_9641_ATP_Olambda__128,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yz,Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_128
tff(fact_9642_ATP_Olambda__129,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yy,Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_129
tff(fact_9643_ATP_Olambda__130,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_uk(Uu),Uua) = aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).

% ATP.lambda_130
tff(fact_9644_ATP_Olambda__131,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_um,Uu),Uua) = aa_real_real(minus_minus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_131
tff(fact_9645_ATP_Olambda__132,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ul(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_132
tff(fact_9646_ATP_Olambda__133,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uo,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_133
tff(fact_9647_ATP_Olambda__134,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_un(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_134
tff(fact_9648_ATP_Olambda__135,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mw,Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_135
tff(fact_9649_ATP_Olambda__136,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mv(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_136
tff(fact_9650_ATP_Olambda__137,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_my,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_137
tff(fact_9651_ATP_Olambda__138,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_mx(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_138
tff(fact_9652_ATP_Olambda__139,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_list_nat(aa_nat6157064778417537467st_nat(aTP_Lamm_wo,Uu),Uua) = cons_nat(Uu,aa_nat_list_nat(nat_list_decode,Uua)) ).

% ATP.lambda_139
tff(fact_9653_ATP_Olambda__140,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_zo,Uu),Uua))
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_140
tff(fact_9654_ATP_Olambda__141,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_pw,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),Uu)) ) ).

% ATP.lambda_141
tff(fact_9655_ATP_Olambda__142,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_px(Uu),Uua))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(Uua),Uu)) ) ).

% ATP.lambda_142
tff(fact_9656_ATP_Olambda__143,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_py,Uu),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Uua),Uu)) ) ).

% ATP.lambda_143
tff(fact_9657_ATP_Olambda__144,axiom,
    ! [Uu: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_pz,Uu),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Uua),Uu)) ) ).

% ATP.lambda_144
tff(fact_9658_ATP_Olambda__145,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ah,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu)) ) ).

% ATP.lambda_145
tff(fact_9659_ATP_Olambda__146,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_qa,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uua),Uu)) ) ).

% ATP.lambda_146
tff(fact_9660_ATP_Olambda__147,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xz,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_147
tff(fact_9661_ATP_Olambda__148,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_li,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),Uu)) ) ).

% ATP.lambda_148
tff(fact_9662_ATP_Olambda__149,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_lh(Uu),Uua))
    <=> pp(aa_set_nat_bool(ord_less_set_nat(Uua),Uu)) ) ).

% ATP.lambda_149
tff(fact_9663_ATP_Olambda__150,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_lj,Uu),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uua),Uu)) ) ).

% ATP.lambda_150
tff(fact_9664_ATP_Olambda__151,axiom,
    ! [Uu: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_lk,Uu),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Uua),Uu)) ) ).

% ATP.lambda_151
tff(fact_9665_ATP_Olambda__152,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_hm,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu)) ) ).

% ATP.lambda_152
tff(fact_9666_ATP_Olambda__153,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ll,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uu)) ) ).

% ATP.lambda_153
tff(fact_9667_ATP_Olambda__154,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yu,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_154
tff(fact_9668_ATP_Olambda__155,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xe,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_155
tff(fact_9669_ATP_Olambda__156,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ys,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_156
tff(fact_9670_ATP_Olambda__157,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xu,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_157
tff(fact_9671_ATP_Olambda__158,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_xk,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_158
tff(fact_9672_ATP_Olambda__159,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_ye(Uu),Uua) = aa_real_real(powr_real(Uua),Uu) ).

% ATP.lambda_159
tff(fact_9673_ATP_Olambda__160,axiom,
    ! [Uu: complex,Uua: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_fo,Uu),Uua))
    <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,Uua),Uu)) ) ).

% ATP.lambda_160
tff(fact_9674_ATP_Olambda__161,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_fp,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,Uua),Uu)) ) ).

% ATP.lambda_161
tff(fact_9675_ATP_Olambda__162,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_fr,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu)) ) ).

% ATP.lambda_162
tff(fact_9676_ATP_Olambda__163,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_fq,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uua),Uu)) ) ).

% ATP.lambda_163
tff(fact_9677_ATP_Olambda__164,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_wx,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_164
tff(fact_9678_ATP_Olambda__165,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_165
tff(fact_9679_ATP_Olambda__166,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_aa(Uu),Uua))
    <=> pp(aa_set_complex_bool(member_complex(Uua),Uu)) ) ).

% ATP.lambda_166
tff(fact_9680_ATP_Olambda__167,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ab(Uu),Uua))
    <=> pp(aa_set_real_bool(member_real(Uua),Uu)) ) ).

% ATP.lambda_167
tff(fact_9681_ATP_Olambda__168,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_ac(Uu),Uua))
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_168
tff(fact_9682_ATP_Olambda__169,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_ad(Uu),Uua))
    <=> member_set_nat(Uua,Uu) ) ).

% ATP.lambda_169
tff(fact_9683_ATP_Olambda__170,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_ae(Uu),Uua))
    <=> pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).

% ATP.lambda_170
tff(fact_9684_ATP_Olambda__171,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_a(Uu),Uua))
    <=> pp(aa_set_int_bool(member_int(Uua),Uu)) ) ).

% ATP.lambda_171
tff(fact_9685_ATP_Olambda__172,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_zx(Uu),Uua) = aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Uu) ).

% ATP.lambda_172
tff(fact_9686_ATP_Olambda__173,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_oj(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_173
tff(fact_9687_ATP_Olambda__174,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_oi(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_174
tff(fact_9688_ATP_Olambda__175,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_kg(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_175
tff(fact_9689_ATP_Olambda__176,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gn(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_176
tff(fact_9690_ATP_Olambda__177,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_wm(Uu),Uua))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_177
tff(fact_9691_ATP_Olambda__178,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_gw(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_178
tff(fact_9692_ATP_Olambda__179,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_gm(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_179
tff(fact_9693_ATP_Olambda__180,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_gx(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_180
tff(fact_9694_ATP_Olambda__181,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_xd,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_xc(Uu),Uua)) ).

% ATP.lambda_181
tff(fact_9695_ATP_Olambda__182,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_xb,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_xa(Uu),Uua)) ).

% ATP.lambda_182
tff(fact_9696_ATP_Olambda__183,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_wz,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_wy(Uu),Uua)) ).

% ATP.lambda_183
tff(fact_9697_ATP_Olambda__184,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_ww,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_wv(Uu),Uua)) ).

% ATP.lambda_184
tff(fact_9698_ATP_Olambda__185,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_yl(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_yk(Uu),Uua)) ).

% ATP.lambda_185
tff(fact_9699_ATP_Olambda__186,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_wl(Uu),Uua) = aa_real_complex(cis,aa_real_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_186
tff(fact_9700_ATP_Olambda__187,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_vc,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_187
tff(fact_9701_ATP_Olambda__188,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_aan(Uu),Uua) = produc6639722614265839536atural(plus_p4538020629002901425atural(Uua,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,Uu),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))))))))))))))))))))))) ).

% ATP.lambda_188
tff(fact_9702_ATP_Olambda__189,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_tp(Uu),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_189
tff(fact_9703_ATP_Olambda__190,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_tr(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_190
tff(fact_9704_ATP_Olambda__191,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_tq(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_191
tff(fact_9705_ATP_Olambda__192,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_kq(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(Uu,aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))) ).

% ATP.lambda_192
tff(fact_9706_ATP_Olambda__193,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_kr(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_193
tff(fact_9707_ATP_Olambda__194,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_kt(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(plus_plus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_194
tff(fact_9708_ATP_Olambda__195,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_ks(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_195
tff(fact_9709_ATP_Olambda__196,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_kv(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_196
tff(fact_9710_ATP_Olambda__197,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_ku(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_197
tff(fact_9711_ATP_Olambda__198,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_zv(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_198
tff(fact_9712_ATP_Olambda__199,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_aap(Uu),Uua) = produc6639722614265839536atural(modulo8411746178871703098atural(Uua,Uu)) ).

% ATP.lambda_199
tff(fact_9713_ATP_Olambda__200,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_aae(Uu),Uua) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uua)) ).

% ATP.lambda_200
tff(fact_9714_ATP_Olambda__201,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zm,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_201
tff(fact_9715_ATP_Olambda__202,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zn,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_202
tff(fact_9716_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ws,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_203
tff(fact_9717_ATP_Olambda__204,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wr,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_204
tff(fact_9718_ATP_Olambda__205,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_yw(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_205
tff(fact_9719_ATP_Olambda__206,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_lm(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uu,Uua)) ).

% ATP.lambda_206
tff(fact_9720_ATP_Olambda__207,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_ln(Uu),Uua) = times_times_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_207
tff(fact_9721_ATP_Olambda__208,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_lo(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_208
tff(fact_9722_ATP_Olambda__209,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_lp(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_209
tff(fact_9723_ATP_Olambda__210,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_lr(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_210
tff(fact_9724_ATP_Olambda__211,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_lq(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_211
tff(fact_9725_ATP_Olambda__212,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_nt(Uu),Uua) = plus_plus_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_212
tff(fact_9726_ATP_Olambda__213,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_nx(Uu),Uua) = plus_plus_real(aa_nat_real(Uu,Uua)) ).

% ATP.lambda_213
tff(fact_9727_ATP_Olambda__214,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_nu(Uu),Uua) = aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_214
tff(fact_9728_ATP_Olambda__215,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_nw(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_215
tff(fact_9729_ATP_Olambda__216,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_nv(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_216
tff(fact_9730_ATP_Olambda__217,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vo(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_217
tff(fact_9731_ATP_Olambda__218,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_vn(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_218
tff(fact_9732_ATP_Olambda__219,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_zd(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_219
tff(fact_9733_ATP_Olambda__220,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_yg(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_220
tff(fact_9734_ATP_Olambda__221,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_nat(aTP_Lamm_iq(Uu),Uua) = aa_nat_nat(suc,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_221
tff(fact_9735_ATP_Olambda__222,axiom,
    ! [Uu: fun_list_nat_nat,Uua: list_nat] : aa_list_nat_nat(aTP_Lamm_is(Uu),Uua) = aa_nat_nat(suc,aa_list_nat_nat(Uu,Uua)) ).

% ATP.lambda_222
tff(fact_9736_ATP_Olambda__223,axiom,
    ! [Uu: fun_set_nat_nat,Uua: set_nat] : aa_set_nat_nat(aTP_Lamm_it(Uu),Uua) = aa_nat_nat(suc,aa_set_nat_nat(Uu,Uua)) ).

% ATP.lambda_223
tff(fact_9737_ATP_Olambda__224,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_iu(Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_224
tff(fact_9738_ATP_Olambda__225,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_nat(aTP_Lamm_ir(Uu),Uua) = aa_nat_nat(suc,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_225
tff(fact_9739_ATP_Olambda__226,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_zw(Uu),Uua) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_hm,Uu)) ).

% ATP.lambda_226
tff(fact_9740_ATP_Olambda__227,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_oh(Uu),Uua),Uub) = aa_real_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_227
tff(fact_9741_ATP_Olambda__228,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_wn(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_228
tff(fact_9742_ATP_Olambda__229,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_wf(Uu),Uua),Uub) = aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(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_229
tff(fact_9743_ATP_Olambda__230,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_wg(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_230
tff(fact_9744_ATP_Olambda__231,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ib(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_set_nat_bool(member_nat(Uub),Uu),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_231
tff(fact_9745_ATP_Olambda__232,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ie(Uu),Uua),Uub) = aa_real_real(if_real(aa_set_nat_bool(member_nat(Uub),Uu),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_232
tff(fact_9746_ATP_Olambda__233,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_id(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_233
tff(fact_9747_ATP_Olambda__234,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ic(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(if_int(aa_set_nat_bool(member_nat(Uub),Uu)),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_234
tff(fact_9748_ATP_Olambda__235,axiom,
    ! [Uu: fun_complex_complex,Uua: set_complex,Uub: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aTP_Lamm_fu(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_set_complex_bool(member_complex(Uub),Uua),aa_complex_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_235
tff(fact_9749_ATP_Olambda__236,axiom,
    ! [Uu: fun_complex_real,Uua: set_complex,Uub: complex] : aa_complex_real(aa_set904117813986905849x_real(aTP_Lamm_fx(Uu),Uua),Uub) = aa_real_real(if_real(aa_set_complex_bool(member_complex(Uub),Uua),aa_complex_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_236
tff(fact_9750_ATP_Olambda__237,axiom,
    ! [Uu: fun_complex_rat,Uua: set_complex,Uub: complex] : aa_complex_rat(aa_set7482584248617139493ex_rat(aTP_Lamm_ga(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_set_complex_bool(member_complex(Uub),Uua)),aa_complex_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_237
tff(fact_9751_ATP_Olambda__238,axiom,
    ! [Uu: fun_real_complex,Uua: set_real,Uub: real] : aa_real_complex(aa_set2225337358267413879omplex(aTP_Lamm_fs(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_set_real_bool(member_real(Uub),Uua),aa_real_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_238
tff(fact_9752_ATP_Olambda__239,axiom,
    ! [Uu: fun_real_real,Uua: set_real,Uub: real] : aa_real_real(aa_set8744029839670122869l_real(aTP_Lamm_fv(Uu),Uua),Uub) = aa_real_real(if_real(aa_set_real_bool(member_real(Uub),Uua),aa_real_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_239
tff(fact_9753_ATP_Olambda__240,axiom,
    ! [Uu: fun_real_rat,Uua: set_real,Uub: real] : aa_real_rat(aa_set4110029769364456865al_rat(aTP_Lamm_fy(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_set_real_bool(member_real(Uub),Uua)),aa_real_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_240
tff(fact_9754_ATP_Olambda__241,axiom,
    ! [Uu: fun_real_nat,Uua: set_real,Uub: real] : aa_real_nat(aa_set7618258327757853593al_nat(aTP_Lamm_gb(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_set_real_bool(member_real(Uub),Uua)),aa_real_nat(Uu,Uub)),zero_zero_nat) ).

% ATP.lambda_241
tff(fact_9755_ATP_Olambda__242,axiom,
    ! [Uu: fun_int_complex,Uua: set_int,Uub: int] : aa_int_complex(aa_set2721330520047917815omplex(aTP_Lamm_ft(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_set_int_bool(member_int(Uub),Uua),aa_int_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_242
tff(fact_9756_ATP_Olambda__243,axiom,
    ! [Uu: fun_int_real,Uua: set_int,Uub: int] : aa_int_real(aa_set6366552415685541621t_real(aTP_Lamm_fw(Uu),Uua),Uub) = aa_real_real(if_real(aa_set_int_bool(member_int(Uub),Uua),aa_int_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_243
tff(fact_9757_ATP_Olambda__244,axiom,
    ! [Uu: fun_int_rat,Uua: set_int,Uub: int] : aa_int_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_fz(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_set_int_bool(member_int(Uub),Uua)),aa_int_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_244
tff(fact_9758_ATP_Olambda__245,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_dr(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub),aa_complex_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_245
tff(fact_9759_ATP_Olambda__246,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_du(Uu),Uua),Uub) = aa_real_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_246
tff(fact_9760_ATP_Olambda__247,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_do(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_247
tff(fact_9761_ATP_Olambda__248,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ds(Uu),Uua),Uub) = aa_real_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_248
tff(fact_9762_ATP_Olambda__249,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_dv(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub)),aa_real_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_249
tff(fact_9763_ATP_Olambda__250,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_dp(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)),zero_zero_complex) ).

% ATP.lambda_250
tff(fact_9764_ATP_Olambda__251,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_dw(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_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_251
tff(fact_9765_ATP_Olambda__252,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_dq(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)),zero_zero_complex) ).

% ATP.lambda_252
tff(fact_9766_ATP_Olambda__253,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_dt(Uu),Uua),Uub) = aa_real_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_253
tff(fact_9767_ATP_Olambda__254,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_dx(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_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_254
tff(fact_9768_ATP_Olambda__255,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_dh(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu),aa_complex_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_255
tff(fact_9769_ATP_Olambda__256,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_dk(Uu),Uua),Uub) = aa_real_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_256
tff(fact_9770_ATP_Olambda__257,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_de(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_257
tff(fact_9771_ATP_Olambda__258,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_di(Uu),Uua),Uub) = aa_real_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_258
tff(fact_9772_ATP_Olambda__259,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_dl(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_259
tff(fact_9773_ATP_Olambda__260,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_df(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)),zero_zero_complex) ).

% ATP.lambda_260
tff(fact_9774_ATP_Olambda__261,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_jz(Uu),Uua),Uub) = aa_real_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_261
tff(fact_9775_ATP_Olambda__262,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_dm(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_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_262
tff(fact_9776_ATP_Olambda__263,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ka(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_263
tff(fact_9777_ATP_Olambda__264,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_kb(Uu),Uua),Uub) = aa_int_int(aa_int_fun_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_264
tff(fact_9778_ATP_Olambda__265,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_dg(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)),zero_zero_complex) ).

% ATP.lambda_265
tff(fact_9779_ATP_Olambda__266,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_dj(Uu),Uua),Uub) = aa_real_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_266
tff(fact_9780_ATP_Olambda__267,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_dn(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_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_267
tff(fact_9781_ATP_Olambda__268,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_zq(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(uminus1351360451143612070nteger(Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(uminus1351360451143612070nteger(Uua),one_one_Code_integer),minus_8373710615458151222nteger(abs_abs_Code_integer(Uu),Uub))) ).

% ATP.lambda_268
tff(fact_9782_ATP_Olambda__269,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_if(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uu,Uub),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_269
tff(fact_9783_ATP_Olambda__270,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ii(Uu),Uua),Uub) = aa_real_real(if_real(aa_nat_bool(Uu,Uub),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_270
tff(fact_9784_ATP_Olambda__271,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ih(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_271
tff(fact_9785_ATP_Olambda__272,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ig(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(if_int(aa_nat_bool(Uu,Uub)),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_272
tff(fact_9786_ATP_Olambda__273,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_bool,Uub: complex] : aa_complex_complex(aa_fun1034237174562290143omplex(aTP_Lamm_fh(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(Uua,Uub),aa_complex_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_273
tff(fact_9787_ATP_Olambda__274,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_fk(Uu),Uua),Uub) = aa_real_real(if_real(aa_complex_bool(Uua,Uub),aa_complex_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_274
tff(fact_9788_ATP_Olambda__275,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_bool,Uub: real] : aa_real_complex(aa_fun6813637976755515871omplex(aTP_Lamm_fb(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(Uua,Uub),aa_real_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_275
tff(fact_9789_ATP_Olambda__276,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_fi(Uu),Uua),Uub) = aa_real_real(if_real(aa_real_bool(Uua,Uub),aa_real_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_276
tff(fact_9790_ATP_Olambda__277,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_bool,Uub: real] : aa_real_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_fl(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_real_bool(Uua,Uub)),aa_real_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_277
tff(fact_9791_ATP_Olambda__278,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_bool,Uub: nat] : aa_nat_complex(aa_fun8588439219086927967omplex(aTP_Lamm_fd(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uua,Uub),aa_nat_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_278
tff(fact_9792_ATP_Olambda__279,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_bool,Uub: nat] : aa_nat_rat(aa_fun5656162108953708681at_rat(aTP_Lamm_fm(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_nat_bool(Uua,Uub)),aa_nat_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_279
tff(fact_9793_ATP_Olambda__280,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_bool,Uub: int] : aa_int_complex(aa_fun3606824691168040799omplex(aTP_Lamm_ff(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(Uua,Uub),aa_int_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_280
tff(fact_9794_ATP_Olambda__281,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_fj(Uu),Uua),Uub) = aa_real_real(if_real(aa_int_bool(Uua,Uub),aa_int_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_281
tff(fact_9795_ATP_Olambda__282,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_bool,Uub: int] : aa_int_rat(aa_fun4400602662665189257nt_rat(aTP_Lamm_fn(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_int_bool(Uua,Uub)),aa_int_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_282
tff(fact_9796_ATP_Olambda__283,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_yf(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_283
tff(fact_9797_ATP_Olambda__284,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_qo(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_284
tff(fact_9798_ATP_Olambda__285,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ql(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_285
tff(fact_9799_ATP_Olambda__286,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qr(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_286
tff(fact_9800_ATP_Olambda__287,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_hy(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_287
tff(fact_9801_ATP_Olambda__288,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),one_one_real),zero_zero_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_288
tff(fact_9802_ATP_Olambda__289,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ia(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_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_289
tff(fact_9803_ATP_Olambda__290,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qs(Uu),Uua),Uub) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qr(Uu),Uub),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_290
tff(fact_9804_ATP_Olambda__291,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qu(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ql(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_291
tff(fact_9805_ATP_Olambda__292,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_qp(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_qo(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_292
tff(fact_9806_ATP_Olambda__293,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qm(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ql(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_293
tff(fact_9807_ATP_Olambda__294,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_xs(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_xr(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_294
tff(fact_9808_ATP_Olambda__295,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_xq(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_xp(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_295
tff(fact_9809_ATP_Olambda__296,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_xo(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_xn(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_296
tff(fact_9810_ATP_Olambda__297,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_xm(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_xl(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_297
tff(fact_9811_ATP_Olambda__298,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_xj(Uu),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_xi(Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_298
tff(fact_9812_ATP_Olambda__299,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_xh(Uu),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_xg(Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_299
tff(fact_9813_ATP_Olambda__300,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ym(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_300
tff(fact_9814_ATP_Olambda__301,axiom,
    ! [Uu: real,Uua: fun_na5723442673449079972x_real,Uub: nat] : aa_nat_real(aa_fun5746040925223427547t_real(aTP_Lamm_ve(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_complex_real(aa_nat480766339200631653x_real(Uua,Uub),zero_zero_complex)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_301
tff(fact_9815_ATP_Olambda__302,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_vf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_302
tff(fact_9816_ATP_Olambda__303,axiom,
    ! [Uu: real,Uua: fun_nat_fun_rat_real,Uub: nat] : aa_nat_real(aa_fun507281639146497029t_real(aTP_Lamm_vg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_rat_real(aa_nat_fun_rat_real(Uua,Uub),zero_zero_rat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_303
tff(fact_9817_ATP_Olambda__304,axiom,
    ! [Uu: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_vh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_nat_real(aa_nat_fun_nat_real(Uua,Uub),zero_zero_nat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_304
tff(fact_9818_ATP_Olambda__305,axiom,
    ! [Uu: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_vi(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_int_real(aa_nat_fun_int_real(Uua,Uub),zero_zero_int)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_305
tff(fact_9819_ATP_Olambda__306,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_qz(Uu),Uua),Uub))
    <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(Uu),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_real ) ) ).

% ATP.lambda_306
tff(fact_9820_ATP_Olambda__307,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_qy(Uu),Uua),Uub))
    <=> ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qi(Uu),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_complex ) ) ).

% ATP.lambda_307
tff(fact_9821_ATP_Olambda__308,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_el(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_308
tff(fact_9822_ATP_Olambda__309,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_en(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_309
tff(fact_9823_ATP_Olambda__310,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_em(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_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_310
tff(fact_9824_ATP_Olambda__311,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_311
tff(fact_9825_ATP_Olambda__312,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_312
tff(fact_9826_ATP_Olambda__313,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_xx(Uu),Uua),Uub))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
        & ( aa_list_nat_nat(groups4561878855575611511st_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_313
tff(fact_9827_ATP_Olambda__314,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_xy(Uu),Uua),Uub))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_nat_nat(groups4561878855575611511st_nat,Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_314
tff(fact_9828_ATP_Olambda__315,axiom,
    ! [Uu: nat,Uua: set_VEBT_VEBT,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_set3008069279726893048T_bool(aTP_Lamm_tl(Uu),Uua),Uub))
    <=> ( ( size_s6755466524823107622T_VEBT(Uub) = Uu )
        & pp(aa_lis2822252094023462681T_bool(distinct_VEBT_VEBT,Uub))
        & ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uua) ) ) ).

% ATP.lambda_315
tff(fact_9829_ATP_Olambda__316,axiom,
    ! [Uu: nat,Uua: set_complex,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_set1925655978115286672x_bool(aTP_Lamm_ti(Uu),Uua),Uub))
    <=> ( ( size_s3451745648224563538omplex(Uub) = Uu )
        & distinct_complex(Uub)
        & pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uua)) ) ) ).

% ATP.lambda_316
tff(fact_9830_ATP_Olambda__317,axiom,
    ! [Uu: nat,Uua: set_list_nat,Uub: list_list_nat] :
      ( pp(aa_lis4609933955469982953t_bool(aa_set3810083887808699692t_bool(aTP_Lamm_tj(Uu),Uua),Uub))
    <=> ( ( size_s3023201423986296836st_nat(Uub) = Uu )
        & distinct_list_nat(Uub)
        & ord_le6045566169113846134st_nat(set_list_nat2(Uub),Uua) ) ) ).

% ATP.lambda_317
tff(fact_9831_ATP_Olambda__318,axiom,
    ! [Uu: nat,Uua: set_bool,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_set1157894234898225080l_bool(aTP_Lamm_tm(Uu),Uua),Uub))
    <=> ( ( size_size_list_bool(Uub) = Uu )
        & pp(aa_list_bool_bool(distinct_bool,Uub))
        & ord_less_eq_set_bool(set_bool2(Uub),Uua) ) ) ).

% ATP.lambda_318
tff(fact_9832_ATP_Olambda__319,axiom,
    ! [Uu: nat,Uua: set_set_nat,Uub: list_set_nat] :
      ( pp(aa_list_set_nat_bool(aa_set736033600575934688t_bool(aTP_Lamm_tk(Uu),Uua),Uub))
    <=> ( ( size_s3254054031482475050et_nat(Uub) = Uu )
        & distinct_set_nat(Uub)
        & ord_le6893508408891458716et_nat(set_set_nat2(Uub),Uua) ) ) ).

% ATP.lambda_319
tff(fact_9833_ATP_Olambda__320,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_set1433770957266526284t_bool(aTP_Lamm_to(Uu),Uua),Uub))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & pp(aa_list_nat_bool(distinct_nat,Uub))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Uub)),Uua)) ) ) ).

% ATP.lambda_320
tff(fact_9834_ATP_Olambda__321,axiom,
    ! [Uu: nat,Uua: set_int,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_set9099777061830656916t_bool(aTP_Lamm_tn(Uu),Uua),Uub))
    <=> ( ( size_size_list_int(Uub) = Uu )
        & pp(aa_list_int_bool(distinct_int,Uub))
        & pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Uub)),Uua)) ) ) ).

% ATP.lambda_321
tff(fact_9835_ATP_Olambda__322,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_ov(Uu),Uua),Uub))
    <=> ( ( size_s6755466524823107622T_VEBT(Uub) = Uua )
        & pp(aa_lis2822252094023462681T_bool(distinct_VEBT_VEBT,Uub))
        & ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu) ) ) ).

% ATP.lambda_322
tff(fact_9836_ATP_Olambda__323,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_ou(Uu),Uua),Uub))
    <=> ( ( size_s3451745648224563538omplex(Uub) = Uua )
        & distinct_complex(Uub)
        & pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)) ) ) ).

% ATP.lambda_323
tff(fact_9837_ATP_Olambda__324,axiom,
    ! [Uu: set_list_nat,Uua: nat,Uub: list_list_nat] :
      ( pp(aa_lis4609933955469982953t_bool(aa_nat3401988846471794802t_bool(aTP_Lamm_os(Uu),Uua),Uub))
    <=> ( ( size_s3023201423986296836st_nat(Uub) = Uua )
        & distinct_list_nat(Uub)
        & ord_le6045566169113846134st_nat(set_list_nat2(Uub),Uu) ) ) ).

% ATP.lambda_324
tff(fact_9838_ATP_Olambda__325,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_ow(Uu),Uua),Uub))
    <=> ( ( size_size_list_bool(Uub) = Uua )
        & pp(aa_list_bool_bool(distinct_bool,Uub))
        & ord_less_eq_set_bool(set_bool2(Uub),Uu) ) ) ).

% ATP.lambda_325
tff(fact_9839_ATP_Olambda__326,axiom,
    ! [Uu: set_set_nat,Uua: nat,Uub: list_set_nat] :
      ( pp(aa_list_set_nat_bool(aa_nat948039967059030092t_bool(aTP_Lamm_ot(Uu),Uua),Uub))
    <=> ( ( size_s3254054031482475050et_nat(Uub) = Uua )
        & distinct_set_nat(Uub)
        & ord_le6893508408891458716et_nat(set_set_nat2(Uub),Uu) ) ) ).

% ATP.lambda_326
tff(fact_9840_ATP_Olambda__327,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_oy(Uu),Uua),Uub))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uua )
        & pp(aa_list_nat_bool(distinct_nat,Uub))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Uub)),Uu)) ) ) ).

% ATP.lambda_327
tff(fact_9841_ATP_Olambda__328,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_ox(Uu),Uua),Uub))
    <=> ( ( size_size_list_int(Uub) = Uua )
        & pp(aa_list_int_bool(distinct_int,Uub))
        & pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Uub)),Uu)) ) ) ).

% ATP.lambda_328
tff(fact_9842_ATP_Olambda__329,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_bf(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_329
tff(fact_9843_ATP_Olambda__330,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_be(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_330
tff(fact_9844_ATP_Olambda__331,axiom,
    ! [Uu: set_list_nat,Uua: nat,Uub: list_list_nat] :
      ( pp(aa_lis4609933955469982953t_bool(aa_nat3401988846471794802t_bool(aTP_Lamm_sy(Uu),Uua),Uub))
    <=> ( ord_le6045566169113846134st_nat(set_list_nat2(Uub),Uu)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s3023201423986296836st_nat(Uub)),Uua)) ) ) ).

% ATP.lambda_331
tff(fact_9845_ATP_Olambda__332,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_bg(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_332
tff(fact_9846_ATP_Olambda__333,axiom,
    ! [Uu: set_set_nat,Uua: nat,Uub: list_set_nat] :
      ( pp(aa_list_set_nat_bool(aa_nat948039967059030092t_bool(aTP_Lamm_sz(Uu),Uua),Uub))
    <=> ( ord_le6893508408891458716et_nat(set_set_nat2(Uub),Uu)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s3254054031482475050et_nat(Uub)),Uua)) ) ) ).

% ATP.lambda_333
tff(fact_9847_ATP_Olambda__334,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_bi(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,aa_list_nat_nat(size_size_list_nat,Uub)),Uua)) ) ) ).

% ATP.lambda_334
tff(fact_9848_ATP_Olambda__335,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_335
tff(fact_9849_ATP_Olambda__336,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_ay(Uu),Uua),Uub))
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_336
tff(fact_9850_ATP_Olambda__337,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_ax(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_337
tff(fact_9851_ATP_Olambda__338,axiom,
    ! [Uu: set_list_nat,Uua: nat,Uub: list_list_nat] :
      ( pp(aa_lis4609933955469982953t_bool(aa_nat3401988846471794802t_bool(aTP_Lamm_bc(Uu),Uua),Uub))
    <=> ( ord_le6045566169113846134st_nat(set_list_nat2(Uub),Uu)
        & ( size_s3023201423986296836st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_338
tff(fact_9852_ATP_Olambda__339,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_az(Uu),Uua),Uub))
    <=> ( ord_less_eq_set_bool(set_bool2(Uub),Uu)
        & ( size_size_list_bool(Uub) = Uua ) ) ) ).

% ATP.lambda_339
tff(fact_9853_ATP_Olambda__340,axiom,
    ! [Uu: set_set_nat,Uua: nat,Uub: list_set_nat] :
      ( pp(aa_list_set_nat_bool(aa_nat948039967059030092t_bool(aTP_Lamm_bd(Uu),Uua),Uub))
    <=> ( ord_le6893508408891458716et_nat(set_set_nat2(Uub),Uu)
        & ( size_s3254054031482475050et_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_340
tff(fact_9854_ATP_Olambda__341,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_bb(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Uub)),Uu))
        & ( aa_list_nat_nat(size_size_list_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_341
tff(fact_9855_ATP_Olambda__342,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_342
tff(fact_9856_ATP_Olambda__343,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_xw(Uu),Uua),Uub))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( aa_list_nat_nat(groups4561878855575611511st_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_343
tff(fact_9857_ATP_Olambda__344,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aw(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_344
tff(fact_9858_ATP_Olambda__345,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_nat8857836526276733660t_bool(aTP_Lamm_aah(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_345
tff(fact_9859_ATP_Olambda__346,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_zr(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_346
tff(fact_9860_ATP_Olambda__347,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_av(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_347
tff(fact_9861_ATP_Olambda__348,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qw(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub)) ).

% ATP.lambda_348
tff(fact_9862_ATP_Olambda__349,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ag(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_349
tff(fact_9863_ATP_Olambda__350,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_af(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_350
tff(fact_9864_ATP_Olambda__351,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_zt(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_351
tff(fact_9865_ATP_Olambda__352,axiom,
    ! [Uu: set_complex,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_fg(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_352
tff(fact_9866_ATP_Olambda__353,axiom,
    ! [Uu: set_real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_fa(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_353
tff(fact_9867_ATP_Olambda__354,axiom,
    ! [Uu: set_nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_fc(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_354
tff(fact_9868_ATP_Olambda__355,axiom,
    ! [Uu: set_int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_fe(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_355
tff(fact_9869_ATP_Olambda__356,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_cn(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_356
tff(fact_9870_ATP_Olambda__357,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_cv(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_357
tff(fact_9871_ATP_Olambda__358,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_ch(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_358
tff(fact_9872_ATP_Olambda__359,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_cp(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_359
tff(fact_9873_ATP_Olambda__360,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aTP_Lamm_cx(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_360
tff(fact_9874_ATP_Olambda__361,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_cj(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_361
tff(fact_9875_ATP_Olambda__362,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_cr(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_362
tff(fact_9876_ATP_Olambda__363,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aTP_Lamm_cz(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_363
tff(fact_9877_ATP_Olambda__364,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_cl(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_364
tff(fact_9878_ATP_Olambda__365,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_ct(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_365
tff(fact_9879_ATP_Olambda__366,axiom,
    ! [Uu: set_complex,Uua: fun_co8153928105989009288nteger,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun6924306962011675644x_bool(aTP_Lamm_bt(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_com4217951221296830281nteger(Uua,Uub) != one_one_Code_integer ) ) ) ).

% ATP.lambda_366
tff(fact_9880_ATP_Olambda__367,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_cb(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_367
tff(fact_9881_ATP_Olambda__368,axiom,
    ! [Uu: set_real,Uua: fun_re6978948838062957830nteger,Uub: real] :
      ( pp(aa_real_bool(aa_fun2607813867624603264l_bool(aTP_Lamm_bn(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_Code_integer(Uua,Uub) != one_one_Code_integer ) ) ) ).

% ATP.lambda_368
tff(fact_9882_ATP_Olambda__369,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_bv(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_369
tff(fact_9883_ATP_Olambda__370,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_cd(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_370
tff(fact_9884_ATP_Olambda__371,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1752071697927428280t_bool(aTP_Lamm_bp(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_Code_integer(Uua,Uub) != one_one_Code_integer ) ) ) ).

% ATP.lambda_371
tff(fact_9885_ATP_Olambda__372,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_bx(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_372
tff(fact_9886_ATP_Olambda__373,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_cf(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_373
tff(fact_9887_ATP_Olambda__374,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: int] :
      ( pp(aa_int_bool(aa_fun3610984093196804608t_bool(aTP_Lamm_br(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_Code_integer(Uua,Uub) != one_one_Code_integer ) ) ) ).

% ATP.lambda_374
tff(fact_9888_ATP_Olambda__375,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_bz(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_375
tff(fact_9889_ATP_Olambda__376,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_zj(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_376
tff(fact_9890_ATP_Olambda__377,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_zu(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_377
tff(fact_9891_ATP_Olambda__378,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_bj(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_378
tff(fact_9892_ATP_Olambda__379,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_bm(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_379
tff(fact_9893_ATP_Olambda__380,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_zc(Uu),Uua),Uub) = aa_real_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_380
tff(fact_9894_ATP_Olambda__381,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yk(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_381
tff(fact_9895_ATP_Olambda__382,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_qi(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_382
tff(fact_9896_ATP_Olambda__383,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_383
tff(fact_9897_ATP_Olambda__384,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_ra(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(Uua),Uub)) ).

% ATP.lambda_384
tff(fact_9898_ATP_Olambda__385,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rs(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_385
tff(fact_9899_ATP_Olambda__386,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_rb(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_386
tff(fact_9900_ATP_Olambda__387,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_at(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_387
tff(fact_9901_ATP_Olambda__388,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_aaj,Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_388
tff(fact_9902_ATP_Olambda__389,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_lg(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_389
tff(fact_9903_ATP_Olambda__390,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_le(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_390
tff(fact_9904_ATP_Olambda__391,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_lf(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_391
tff(fact_9905_ATP_Olambda__392,axiom,
    ! [Uu: fun_co8153928105989009288nteger,Uua: fun_co8153928105989009288nteger,Uub: complex] : aa_com4217951221296830281nteger(aa_fun5267417455630836558nteger(aTP_Lamm_vp(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_com4217951221296830281nteger(Uua,Uub)),aa_com4217951221296830281nteger(Uu,Uub)) ).

% ATP.lambda_392
tff(fact_9906_ATP_Olambda__393,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_vt(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uub)),aa_complex_real(Uu,Uub)) ).

% ATP.lambda_393
tff(fact_9907_ATP_Olambda__394,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_vw(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_complex_rat(Uua,Uub)),aa_complex_rat(Uu,Uub)) ).

% ATP.lambda_394
tff(fact_9908_ATP_Olambda__395,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_vq(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu,Uub)) ).

% ATP.lambda_395
tff(fact_9909_ATP_Olambda__396,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vu(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_396
tff(fact_9910_ATP_Olambda__397,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_vx(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(Uua,Uub)),aa_real_rat(Uu,Uub)) ).

% ATP.lambda_397
tff(fact_9911_ATP_Olambda__398,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_vr(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_398
tff(fact_9912_ATP_Olambda__399,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_vy(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_399
tff(fact_9913_ATP_Olambda__400,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_vs(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_400
tff(fact_9914_ATP_Olambda__401,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_vv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).

% ATP.lambda_401
tff(fact_9915_ATP_Olambda__402,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_yv(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_402
tff(fact_9916_ATP_Olambda__403,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_il(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_403
tff(fact_9917_ATP_Olambda__404,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_im(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_404
tff(fact_9918_ATP_Olambda__405,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_in(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_405
tff(fact_9919_ATP_Olambda__406,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ip(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_406
tff(fact_9920_ATP_Olambda__407,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_io(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_407
tff(fact_9921_ATP_Olambda__408,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yi(Uu),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_408
tff(fact_9922_ATP_Olambda__409,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_pe(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uu,Uub)),zero_n3304061248610475627l_real(aa_complex_bool(Uua,Uub))) ).

% ATP.lambda_409
tff(fact_9923_ATP_Olambda__410,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_bool,Uub: complex] : aa_complex_rat(aa_fun5529999419959482889ex_rat(aTP_Lamm_ph(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_complex_rat(Uu,Uub)),zero_n2052037380579107095ol_rat(aa_complex_bool(Uua,Uub))) ).

% ATP.lambda_410
tff(fact_9924_ATP_Olambda__411,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_bool,Uub: complex] : aa_complex_nat(aa_fun9038227978352879617ex_nat(aTP_Lamm_pl(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_complex_nat(Uu,Uub)),zero_n2687167440665602831ol_nat(aa_complex_bool(Uua,Uub))) ).

% ATP.lambda_411
tff(fact_9925_ATP_Olambda__412,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_pc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uu,Uub)),zero_n3304061248610475627l_real(aa_real_bool(Uua,Uub))) ).

% ATP.lambda_412
tff(fact_9926_ATP_Olambda__413,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_bool,Uub: real] : aa_real_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_pf(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(Uu,Uub)),zero_n2052037380579107095ol_rat(aa_real_bool(Uua,Uub))) ).

% ATP.lambda_413
tff(fact_9927_ATP_Olambda__414,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_bool,Uub: real] : aa_real_nat(aa_fun1504556115071737857al_nat(aTP_Lamm_pj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_real_nat(Uu,Uub)),zero_n2687167440665602831ol_nat(aa_real_bool(Uua,Uub))) ).

% ATP.lambda_414
tff(fact_9928_ATP_Olambda__415,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_bool,Uub: nat] : aa_nat_rat(aa_fun5656162108953708681at_rat(aTP_Lamm_pi(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uub)),zero_n2052037380579107095ol_rat(aa_nat_bool(Uua,Uub))) ).

% ATP.lambda_415
tff(fact_9929_ATP_Olambda__416,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_pd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uu,Uub)),zero_n3304061248610475627l_real(aa_int_bool(Uua,Uub))) ).

% ATP.lambda_416
tff(fact_9930_ATP_Olambda__417,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_bool,Uub: int] : aa_int_rat(aa_fun4400602662665189257nt_rat(aTP_Lamm_pg(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(Uu,Uub)),zero_n2052037380579107095ol_rat(aa_int_bool(Uua,Uub))) ).

% ATP.lambda_417
tff(fact_9931_ATP_Olambda__418,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_bool,Uub: int] : aa_int_nat(aa_fun7908831221058585985nt_nat(aTP_Lamm_pk(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(Uu,Uub)),zero_n2687167440665602831ol_nat(aa_int_bool(Uua,Uub))) ).

% ATP.lambda_418
tff(fact_9932_ATP_Olambda__419,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_au(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua)) ) ).

% ATP.lambda_419
tff(fact_9933_ATP_Olambda__420,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_et(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_420
tff(fact_9934_ATP_Olambda__421,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_er(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_421
tff(fact_9935_ATP_Olambda__422,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_es(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_422
tff(fact_9936_ATP_Olambda__423,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ke(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uub)),Uu) ).

% ATP.lambda_423
tff(fact_9937_ATP_Olambda__424,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_kf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_424
tff(fact_9938_ATP_Olambda__425,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ma(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_425
tff(fact_9939_ATP_Olambda__426,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_ly(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu,Uub)),Uua) ).

% ATP.lambda_426
tff(fact_9940_ATP_Olambda__427,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_lz(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_427
tff(fact_9941_ATP_Olambda__428,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_yd(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_428
tff(fact_9942_ATP_Olambda__429,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zh(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_429
tff(fact_9943_ATP_Olambda__430,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_yh(Uu),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_430
tff(fact_9944_ATP_Olambda__431,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_po(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_n3304061248610475627l_real(aa_complex_bool(Uu,Uub))),aa_complex_real(Uua,Uub)) ).

% ATP.lambda_431
tff(fact_9945_ATP_Olambda__432,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_pr(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_n2052037380579107095ol_rat(aa_complex_bool(Uu,Uub))),aa_complex_rat(Uua,Uub)) ).

% ATP.lambda_432
tff(fact_9946_ATP_Olambda__433,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_pv(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_complex_bool(Uu,Uub))),aa_complex_nat(Uua,Uub)) ).

% ATP.lambda_433
tff(fact_9947_ATP_Olambda__434,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_pm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_n3304061248610475627l_real(aa_real_bool(Uu,Uub))),aa_real_real(Uua,Uub)) ).

% ATP.lambda_434
tff(fact_9948_ATP_Olambda__435,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_pp(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_n2052037380579107095ol_rat(aa_real_bool(Uu,Uub))),aa_real_rat(Uua,Uub)) ).

% ATP.lambda_435
tff(fact_9949_ATP_Olambda__436,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_pt(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_real_bool(Uu,Uub))),aa_real_nat(Uua,Uub)) ).

% ATP.lambda_436
tff(fact_9950_ATP_Olambda__437,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ps(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_n2052037380579107095ol_rat(aa_nat_bool(Uu,Uub))),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_437
tff(fact_9951_ATP_Olambda__438,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_pn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_n3304061248610475627l_real(aa_int_bool(Uu,Uub))),aa_int_real(Uua,Uub)) ).

% ATP.lambda_438
tff(fact_9952_ATP_Olambda__439,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_pq(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_n2052037380579107095ol_rat(aa_int_bool(Uu,Uub))),aa_int_rat(Uua,Uub)) ).

% ATP.lambda_439
tff(fact_9953_ATP_Olambda__440,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_pu(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_int_bool(Uu,Uub))),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_440
tff(fact_9954_ATP_Olambda__441,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_hn(Uu),Uua),Uub) = plus_p5714425477246183910nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Uub)),Uua)) ).

% ATP.lambda_441
tff(fact_9955_ATP_Olambda__442,axiom,
    ! [Uu: extended_enat,Uua: extended_enat,Uub: nat] : aa_nat_Extended_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_hp(Uu),Uua),Uub) = plus_p3455044024723400733d_enat(Uu,times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uub),Uua)) ).

% ATP.lambda_442
tff(fact_9956_ATP_Olambda__443,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_hq(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_443
tff(fact_9957_ATP_Olambda__444,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ht(Uu),Uua),Uub) = aa_real_real(plus_plus_real(Uu),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_444
tff(fact_9958_ATP_Olambda__445,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_ho(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_445
tff(fact_9959_ATP_Olambda__446,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hs(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_446
tff(fact_9960_ATP_Olambda__447,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_hr(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_447
tff(fact_9961_ATP_Olambda__448,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_oc(Uu),Uua),Uub) = aa_nat_Code_integer(power_8256067586552552935nteger(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_448
tff(fact_9962_ATP_Olambda__449,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_od(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_449
tff(fact_9963_ATP_Olambda__450,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_og(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_450
tff(fact_9964_ATP_Olambda__451,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_oe(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_451
tff(fact_9965_ATP_Olambda__452,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_of(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_452
tff(fact_9966_ATP_Olambda__453,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_gq(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_453
tff(fact_9967_ATP_Olambda__454,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_gt(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_454
tff(fact_9968_ATP_Olambda__455,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_gr(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_455
tff(fact_9969_ATP_Olambda__456,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_gs(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_456
tff(fact_9970_ATP_Olambda__457,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hu(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_457
tff(fact_9971_ATP_Olambda__458,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_kd(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_458
tff(fact_9972_ATP_Olambda__459,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_eq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_459
tff(fact_9973_ATP_Olambda__460,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_eo(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_460
tff(fact_9974_ATP_Olambda__461,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ep(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,Uu),aa_int_int(Uua,Uub)) ).

% ATP.lambda_461
tff(fact_9975_ATP_Olambda__462,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_462
tff(fact_9976_ATP_Olambda__463,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_nz(Uu),Uua),Uub) = aa_real_real(plus_plus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_463
tff(fact_9977_ATP_Olambda__464,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_ny(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_464
tff(fact_9978_ATP_Olambda__465,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ob(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_465
tff(fact_9979_ATP_Olambda__466,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_oa(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_466
tff(fact_9980_ATP_Olambda__467,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_lu(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_467
tff(fact_9981_ATP_Olambda__468,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ld(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_468
tff(fact_9982_ATP_Olambda__469,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_lc(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_469
tff(fact_9983_ATP_Olambda__470,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_gp(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_470
tff(fact_9984_ATP_Olambda__471,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_go(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_471
tff(fact_9985_ATP_Olambda__472,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ls(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_472
tff(fact_9986_ATP_Olambda__473,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ij(Uu),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_473
tff(fact_9987_ATP_Olambda__474,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ik(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_474
tff(fact_9988_ATP_Olambda__475,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zb(Uu),Uua),Uub) = aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),Uua)) ).

% ATP.lambda_475
tff(fact_9989_ATP_Olambda__476,axiom,
    ! [Uu: set_nat,Uua: fun_co2841405903858895691t_bool,Uub: complex] : aa_complex_nat(aa_fun9050545535516344074ex_nat(aTP_Lamm_jy(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_jx(Uu),Uua),Uub))) ).

% ATP.lambda_476
tff(fact_9990_ATP_Olambda__477,axiom,
    ! [Uu: set_real,Uua: fun_co2286642472385812463l_bool,Uub: complex] : aa_complex_nat(aa_fun2409747231911459374ex_nat(aTP_Lamm_jv(Uu),Uua),Uub) = finite_card_real(collect_real(aa_com8943241093633526360l_bool(aa_fun4686908292940058492l_bool(aTP_Lamm_ju(Uu),Uua),Uub))) ).

% ATP.lambda_477
tff(fact_9991_ATP_Olambda__478,axiom,
    ! [Uu: set_complex,Uua: fun_in5750378413879446255x_bool,Uub: int] : aa_int_nat(aa_fun8429035501195100204nt_nat(aTP_Lamm_js(Uu),Uua),Uub) = finite_card_complex(collect_complex(aa_int5354780257279585880x_bool(aa_fun6456964791495427196x_bool(aTP_Lamm_jr(Uu),Uua),Uub))) ).

% ATP.lambda_478
tff(fact_9992_ATP_Olambda__479,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_bool,Uub: int] : aa_int_nat(aa_fun8170219554457397550nt_nat(aTP_Lamm_jp(Uu),Uua),Uub) = finite_card_int(collect_int(aa_int_fun_int_bool(aa_fun8168818243683596540t_bool(aTP_Lamm_jo(Uu),Uua),Uub))) ).

% ATP.lambda_479
tff(fact_9993_ATP_Olambda__480,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_bool,Uub: int] : aa_int_nat(aa_fun5132377469993263754nt_nat(aTP_Lamm_jm(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_jl(Uu),Uua),Uub))) ).

% ATP.lambda_480
tff(fact_9994_ATP_Olambda__481,axiom,
    ! [Uu: set_real,Uua: fun_in3970322703178350321l_bool,Uub: int] : aa_int_nat(aa_fun91123856807183534nt_nat(aTP_Lamm_jj(Uu),Uua),Uub) = finite_card_real(collect_real(aa_int_fun_real_bool(aa_fun5541803710083441916l_bool(aTP_Lamm_ji(Uu),Uua),Uub))) ).

% ATP.lambda_481
tff(fact_9995_ATP_Olambda__482,axiom,
    ! [Uu: set_complex,Uua: fun_re3010880487650781423x_bool,Uub: real] : aa_real_nat(aa_fun5167920629197714732al_nat(aTP_Lamm_jg(Uu),Uua),Uub) = finite_card_complex(collect_complex(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_jf(Uu),Uua),Uub))) ).

% ATP.lambda_482
tff(fact_9996_ATP_Olambda__483,axiom,
    ! [Uu: set_int,Uua: fun_re8661203372455305201t_bool,Uub: real] : aa_real_nat(aa_fun4707068664741690670al_nat(aTP_Lamm_jd(Uu),Uua),Uub) = finite_card_int(collect_int(aa_real_fun_int_bool(aa_fun6227302519962330364t_bool(aTP_Lamm_jc(Uu),Uua),Uub))) ).

% ATP.lambda_483
tff(fact_9997_ATP_Olambda__484,axiom,
    ! [Uu: set_nat,Uua: fun_re9121486324101951821t_bool,Uub: real] : aa_real_nat(aa_fun5804151645385689226al_nat(aTP_Lamm_ja(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_iz(Uu),Uua),Uub))) ).

% ATP.lambda_484
tff(fact_9998_ATP_Olambda__485,axiom,
    ! [Uu: set_real,Uua: fun_re1255401033344146673l_bool,Uub: real] : aa_real_nat(aa_fun3074920444450379950al_nat(aTP_Lamm_ix(Uu),Uua),Uub) = finite_card_real(collect_real(aa_rea7809428808904475482l_bool(aa_fun1425803159125577980l_bool(aTP_Lamm_iw(Uu),Uua),Uub))) ).

% ATP.lambda_485
tff(fact_9999_ATP_Olambda__486,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_rw(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),zero_zero_complex),aa_nat_complex(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_486
tff(fact_10000_ATP_Olambda__487,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_se(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_real(Uua,Uuc)),aa_real_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_487
tff(fact_10001_ATP_Olambda__488,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ry(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_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(aa_rat_fun_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_488
tff(fact_10002_ATP_Olambda__489,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sc(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_489
tff(fact_10003_ATP_Olambda__490,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_sa(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_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(aa_int_fun_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_490
tff(fact_10004_ATP_Olambda__491,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: fun_nat_Code_integer,Uuc: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_sg(Uu),Uua),Uub),Uuc) = if_Code_integer(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_Code_integer(Uua,Uuc),if_Code_integer(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_Code_integer,aa_nat_Code_integer(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_491
tff(fact_10005_ATP_Olambda__492,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_si(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_492
tff(fact_10006_ATP_Olambda__493,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_sj(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_real(Uua,Uuc)),aa_real_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_493
tff(fact_10007_ATP_Olambda__494,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sl(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_494
tff(fact_10008_ATP_Olambda__495,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_sk(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_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(aa_int_fun_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_495
tff(fact_10009_ATP_Olambda__496,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: fun_nat_Code_integer,Uuc: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_sh(Uu),Uua),Uub),Uuc) = if_Code_integer(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_Code_integer(Uua,Uuc),aa_nat_Code_integer(Uub,Uuc)) ).

% ATP.lambda_496
tff(fact_10010_ATP_Olambda__497,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_rx(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_497
tff(fact_10011_ATP_Olambda__498,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_sf(Uu),Uua),Uub),Uuc) = aa_real_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_498
tff(fact_10012_ATP_Olambda__499,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rz(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_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_499
tff(fact_10013_ATP_Olambda__500,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sd(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_500
tff(fact_10014_ATP_Olambda__501,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_sb(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_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_501
tff(fact_10015_ATP_Olambda__502,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_mb(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu),aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_502
tff(fact_10016_ATP_Olambda__503,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_mf(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu)),aa_complex_rat(Uua,Uuc)),aa_complex_rat(Uub,Uuc)) ).

% ATP.lambda_503
tff(fact_10017_ATP_Olambda__504,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_mj(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu)),aa_complex_nat(Uua,Uuc)),aa_complex_nat(Uub,Uuc)) ).

% ATP.lambda_504
tff(fact_10018_ATP_Olambda__505,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] : aa_real_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_md(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu),aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) ).

% ATP.lambda_505
tff(fact_10019_ATP_Olambda__506,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_mh(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu)),aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) ).

% ATP.lambda_506
tff(fact_10020_ATP_Olambda__507,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_me(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_507
tff(fact_10021_ATP_Olambda__508,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_mi(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_508
tff(fact_10022_ATP_Olambda__509,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] : aa_int_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_mc(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),Uu),aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) ).

% ATP.lambda_509
tff(fact_10023_ATP_Olambda__510,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_mg(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),Uu)),aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) ).

% ATP.lambda_510
tff(fact_10024_ATP_Olambda__511,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_mk(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),Uu)),aa_int_nat(Uua,Uuc)),aa_int_nat(Uub,Uuc)) ).

% ATP.lambda_511
tff(fact_10025_ATP_Olambda__512,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex,Uuc: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_mo(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu),aa_complex_complex(Uua,Uuc)),Uub) ).

% ATP.lambda_512
tff(fact_10026_ATP_Olambda__513,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: real,Uuc: complex] : aa_complex_real(aa_rea4562580341872191937x_real(aa_fun3210294639341725717x_real(aTP_Lamm_ms(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu),aa_complex_real(Uua,Uuc)),Uub) ).

% ATP.lambda_513
tff(fact_10027_ATP_Olambda__514,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: complex,Uuc: real] : aa_real_complex(aa_com4766338864777664963omplex(aa_fun3608241143110160151omplex(aTP_Lamm_ml(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu),aa_real_complex(Uua,Uuc)),Uub) ).

% ATP.lambda_514
tff(fact_10028_ATP_Olambda__515,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real,Uuc: real] : aa_real_real(aa_rea7340605660300375871l_real(aa_fun4910002773489250577l_real(aTP_Lamm_mp(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu),aa_real_real(Uua,Uuc)),Uub) ).

% ATP.lambda_515
tff(fact_10029_ATP_Olambda__516,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: rat,Uuc: real] : aa_real_rat(aa_rat_fun_real_rat(aa_fun2328011964990066069al_rat(aTP_Lamm_mt(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uuc),Uu)),aa_real_rat(Uua,Uuc)),Uub) ).

% ATP.lambda_516
tff(fact_10030_ATP_Olambda__517,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_mm(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc)),Uub) ).

% ATP.lambda_517
tff(fact_10031_ATP_Olambda__518,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_mq(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),aa_nat_real(Uua,Uuc)),Uub) ).

% ATP.lambda_518
tff(fact_10032_ATP_Olambda__519,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_mu(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),aa_nat_rat(Uua,Uuc)),Uub) ).

% ATP.lambda_519
tff(fact_10033_ATP_Olambda__520,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: complex,Uuc: int] : aa_int_complex(aa_com9156536935073639875omplex(aa_fun4786838533949490199omplex(aTP_Lamm_mn(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),Uu),aa_int_complex(Uua,Uuc)),Uub) ).

% ATP.lambda_520
tff(fact_10034_ATP_Olambda__521,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: real,Uuc: int] : aa_int_real(aa_real_fun_int_real(aa_fun8936210173152961553t_real(aTP_Lamm_mr(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),Uu),aa_int_real(Uua,Uuc)),Uub) ).

% ATP.lambda_521
tff(fact_10035_ATP_Olambda__522,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_tz(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_complex_bool(Uu,Uuc),aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_522
tff(fact_10036_ATP_Olambda__523,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_ud(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_complex_bool(Uu,Uuc)),aa_complex_rat(Uua,Uuc)),aa_complex_rat(Uub,Uuc)) ).

% ATP.lambda_523
tff(fact_10037_ATP_Olambda__524,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] : aa_real_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_tx(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_real_bool(Uu,Uuc),aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) ).

% ATP.lambda_524
tff(fact_10038_ATP_Olambda__525,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_ub(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_real_bool(Uu,Uuc)),aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) ).

% ATP.lambda_525
tff(fact_10039_ATP_Olambda__526,axiom,
    ! [Uu: fun_real_bool,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: real] : aa_real_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_uf(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(Uu,Uuc)),aa_real_nat(Uua,Uuc)),aa_real_nat(Uub,Uuc)) ).

% ATP.lambda_526
tff(fact_10040_ATP_Olambda__527,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_ua(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_nat_bool(Uu,Uuc),aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_527
tff(fact_10041_ATP_Olambda__528,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ue(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_nat_bool(Uu,Uuc)),aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_528
tff(fact_10042_ATP_Olambda__529,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] : aa_int_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_ty(Uu),Uua),Uub),Uuc) = aa_real_real(if_real(aa_int_bool(Uu,Uuc),aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) ).

% ATP.lambda_529
tff(fact_10043_ATP_Olambda__530,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_uc(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(if_rat(aa_int_bool(Uu,Uuc)),aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) ).

% ATP.lambda_530
tff(fact_10044_ATP_Olambda__531,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_ug(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(Uu,Uuc)),aa_int_nat(Uua,Uuc)),aa_int_nat(Uub,Uuc)) ).

% ATP.lambda_531
tff(fact_10045_ATP_Olambda__532,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_kp(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ko(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_532
tff(fact_10046_ATP_Olambda__533,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_kn(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_km(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_533
tff(fact_10047_ATP_Olambda__534,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_kl(Uu),Uua),Uub),Uuc) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_kk(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_534
tff(fact_10048_ATP_Olambda__535,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: set_nat,Uuc: nat] : aa_nat_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_ez(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_ey(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_535
tff(fact_10049_ATP_Olambda__536,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: set_int,Uuc: int] : aa_int_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_ex(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_ew(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_536
tff(fact_10050_ATP_Olambda__537,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: set_nat,Uuc: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_ev(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_eu(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_537
tff(fact_10051_ATP_Olambda__538,axiom,
    ! [Uu: nat,Uua: code_integer,Uub: code_integer,Uuc: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_uu(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),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_538
tff(fact_10052_ATP_Olambda__539,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_ut(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_539
tff(fact_10053_ATP_Olambda__540,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_uw(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_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(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_real),zero_zero_real))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uuc)) ).

% ATP.lambda_540
tff(fact_10054_ATP_Olambda__541,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_uv(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_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(aa_rat_fun_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_541
tff(fact_10055_ATP_Olambda__542,axiom,
    ! [Uu: nat,Uua: int,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_us(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_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(aa_int_fun_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_542
tff(fact_10056_ATP_Olambda__543,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_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_rt(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_543
tff(fact_10057_ATP_Olambda__544,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_rk(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_rj(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_544
tff(fact_10058_ATP_Olambda__545,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_ri(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_rh(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_int(power_power_int(Uub),Uuc)) ).

% ATP.lambda_545
tff(fact_10059_ATP_Olambda__546,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_rg(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_rf(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_rat(power_power_rat(Uub),Uuc)) ).

% ATP.lambda_546
tff(fact_10060_ATP_Olambda__547,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_re(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_rd(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_547
tff(fact_10061_ATP_Olambda__548,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_yr(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_548
tff(fact_10062_ATP_Olambda__549,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_yp(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_549
tff(fact_10063_ATP_Olambda__550,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yo(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uub),Uua)),Uuc)) ).

% ATP.lambda_550
tff(fact_10064_ATP_Olambda__551,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yn(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uua),Uub)),Uuc)) ).

% ATP.lambda_551
tff(fact_10065_ATP_Olambda__552,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_mz(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_552
tff(fact_10066_ATP_Olambda__553,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nf(Uu),Uua),Uub),Uuc) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_553
tff(fact_10067_ATP_Olambda__554,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nb(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,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_554
tff(fact_10068_ATP_Olambda__555,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_nd(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_555
tff(fact_10069_ATP_Olambda__556,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_wv(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_556
tff(fact_10070_ATP_Olambda__557,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_kk(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_557
tff(fact_10071_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_ko(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Uua),Uu)),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_558
tff(fact_10072_ATP_Olambda__559,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_km(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uua),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_559
tff(fact_10073_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_rr(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(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_560
tff(fact_10074_ATP_Olambda__561,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rp(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uu),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_561
tff(fact_10075_ATP_Olambda__562,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rq(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_562
tff(fact_10076_ATP_Olambda__563,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_563
tff(fact_10077_ATP_Olambda__564,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(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_564
tff(fact_10078_ATP_Olambda__565,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(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc))),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_565
tff(fact_10079_ATP_Olambda__566,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_rc(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_566
tff(fact_10080_ATP_Olambda__567,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_567
tff(fact_10081_ATP_Olambda__568,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nl(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_568
tff(fact_10082_ATP_Olambda__569,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_no(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)) ).

% ATP.lambda_569
tff(fact_10083_ATP_Olambda__570,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_nm(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_rat(power_power_rat(Uu),Uuc)) ).

% ATP.lambda_570
tff(fact_10084_ATP_Olambda__571,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_nn(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_571
tff(fact_10085_ATP_Olambda__572,axiom,
    ! [Uu: set_complex,Uua: fun_re3010880487650781423x_bool,Uub: real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_jf(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_572
tff(fact_10086_ATP_Olambda__573,axiom,
    ! [Uu: set_complex,Uua: fun_in5750378413879446255x_bool,Uub: int,Uuc: complex] :
      ( pp(aa_complex_bool(aa_int5354780257279585880x_bool(aa_fun6456964791495427196x_bool(aTP_Lamm_jr(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_int5354780257279585880x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_573
tff(fact_10087_ATP_Olambda__574,axiom,
    ! [Uu: set_real,Uua: fun_co2286642472385812463l_bool,Uub: complex,Uuc: real] :
      ( pp(aa_real_bool(aa_com8943241093633526360l_bool(aa_fun4686908292940058492l_bool(aTP_Lamm_ju(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_real_bool(aa_com8943241093633526360l_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_574
tff(fact_10088_ATP_Olambda__575,axiom,
    ! [Uu: set_real,Uua: fun_re1255401033344146673l_bool,Uub: real,Uuc: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aa_fun1425803159125577980l_bool(aTP_Lamm_iw(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_575
tff(fact_10089_ATP_Olambda__576,axiom,
    ! [Uu: set_real,Uua: fun_in3970322703178350321l_bool,Uub: int,Uuc: real] :
      ( pp(aa_real_bool(aa_int_fun_real_bool(aa_fun5541803710083441916l_bool(aTP_Lamm_ji(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_real_bool(aa_int_fun_real_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_576
tff(fact_10090_ATP_Olambda__577,axiom,
    ! [Uu: set_nat,Uua: fun_co2841405903858895691t_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_jx(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_com5335841084339320244t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_577
tff(fact_10091_ATP_Olambda__578,axiom,
    ! [Uu: set_nat,Uua: fun_re9121486324101951821t_bool,Uub: real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_iz(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_real_fun_nat_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_578
tff(fact_10092_ATP_Olambda__579,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_bool,Uub: int,Uuc: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_jl(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_579
tff(fact_10093_ATP_Olambda__580,axiom,
    ! [Uu: set_int,Uua: fun_re8661203372455305201t_bool,Uub: real,Uuc: int] :
      ( pp(aa_int_bool(aa_real_fun_int_bool(aa_fun6227302519962330364t_bool(aTP_Lamm_jc(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_real_fun_int_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_580
tff(fact_10094_ATP_Olambda__581,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_bool,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_fun8168818243683596540t_bool(aTP_Lamm_jo(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_int_fun_int_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_581
tff(fact_10095_ATP_Olambda__582,axiom,
    ! [Uu: set_complex,Uua: fun_co2286642472385812463l_bool,Uub: real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_rea5887071448072802392x_bool(aa_fun5411146308205027452x_bool(aTP_Lamm_jt(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_real_bool(aa_com8943241093633526360l_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_582
tff(fact_10096_ATP_Olambda__583,axiom,
    ! [Uu: set_complex,Uua: fun_co2841405903858895691t_bool,Uub: nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_jw(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_nat_bool(aa_com5335841084339320244t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_583
tff(fact_10097_ATP_Olambda__584,axiom,
    ! [Uu: set_real,Uua: fun_re3010880487650781423x_bool,Uub: complex,Uuc: real] :
      ( pp(aa_real_bool(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_je(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_584
tff(fact_10098_ATP_Olambda__585,axiom,
    ! [Uu: set_real,Uua: fun_re1255401033344146673l_bool,Uub: real,Uuc: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aa_fun1425803159125577980l_bool(aTP_Lamm_iv(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_585
tff(fact_10099_ATP_Olambda__586,axiom,
    ! [Uu: set_real,Uua: fun_re9121486324101951821t_bool,Uub: nat,Uuc: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_iy(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_nat_bool(aa_real_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_586
tff(fact_10100_ATP_Olambda__587,axiom,
    ! [Uu: set_real,Uua: fun_re8661203372455305201t_bool,Uub: int,Uuc: real] :
      ( pp(aa_real_bool(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_jb(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_int_bool(aa_real_fun_int_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_587
tff(fact_10101_ATP_Olambda__588,axiom,
    ! [Uu: set_int,Uua: fun_in5750378413879446255x_bool,Uub: complex,Uuc: int] :
      ( pp(aa_int_bool(aa_com2211550117870696024t_bool(aa_fun3087709329828230012t_bool(aTP_Lamm_jq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_complex_bool(aa_int5354780257279585880x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_588
tff(fact_10102_ATP_Olambda__589,axiom,
    ! [Uu: set_int,Uua: fun_in3970322703178350321l_bool,Uub: real,Uuc: int] :
      ( pp(aa_int_bool(aa_real_fun_int_bool(aa_fun1009312342505620988t_bool(aTP_Lamm_jh(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_real_bool(aa_int_fun_real_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_589
tff(fact_10103_ATP_Olambda__590,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_bool,Uub: nat,Uuc: int] :
      ( pp(aa_int_bool(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_jk(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_590
tff(fact_10104_ATP_Olambda__591,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_bool,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_fun8168818243683596540t_bool(aTP_Lamm_jn(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_int_bool(aa_int_fun_int_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_591
tff(fact_10105_ATP_Olambda__592,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_na(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_592
tff(fact_10106_ATP_Olambda__593,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ng(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_593
tff(fact_10107_ATP_Olambda__594,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nc(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Uu),Uuc)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_594
tff(fact_10108_ATP_Olambda__595,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ne(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_595
tff(fact_10109_ATP_Olambda__596,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_nh(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_596
tff(fact_10110_ATP_Olambda__597,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_nk(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_597
tff(fact_10111_ATP_Olambda__598,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ni(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_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_598
tff(fact_10112_ATP_Olambda__599,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_nj(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_599
tff(fact_10113_ATP_Olambda__600,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qx(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_600
tff(fact_10114_ATP_Olambda__601,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_xg(Uu),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ) ).

% ATP.lambda_601
tff(fact_10115_ATP_Olambda__602,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_wy(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_602
tff(fact_10116_ATP_Olambda__603,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_xi(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_603
tff(fact_10117_ATP_Olambda__604,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_xa(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_604
tff(fact_10118_ATP_Olambda__605,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_xc(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_605
tff(fact_10119_ATP_Olambda__606,axiom,
    ! [Uu: set_complex,Uua: fun_co8153928105989009288nteger,Uub: fun_co8153928105989009288nteger,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun6924306962011675644x_bool(aa_fun4664215553580952141x_bool(aTP_Lamm_bu(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_com4217951221296830281nteger(Uua,Uuc)),aa_com4217951221296830281nteger(Uub,Uuc)) != one_one_Code_integer ) ) ) ).

% ATP.lambda_606
tff(fact_10120_ATP_Olambda__607,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_cc(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_607
tff(fact_10121_ATP_Olambda__608,axiom,
    ! [Uu: set_real,Uua: fun_re6978948838062957830nteger,Uub: fun_re6978948838062957830nteger,Uuc: real] :
      ( pp(aa_real_bool(aa_fun2607813867624603264l_bool(aa_fun1301660912713367635l_bool(aTP_Lamm_bo(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uuc)),aa_real_Code_integer(Uub,Uuc)) != one_one_Code_integer ) ) ) ).

% ATP.lambda_608
tff(fact_10122_ATP_Olambda__609,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_bw(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(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_609
tff(fact_10123_ATP_Olambda__610,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_ce(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(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_610
tff(fact_10124_ATP_Olambda__611,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: fun_nat_Code_integer,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1752071697927428280t_bool(aa_fun4813323487062687079t_bool(aTP_Lamm_bq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uuc)),aa_nat_Code_integer(Uub,Uuc)) != one_one_Code_integer ) ) ) ).

% ATP.lambda_611
tff(fact_10125_ATP_Olambda__612,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_by(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_612
tff(fact_10126_ATP_Olambda__613,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_cg(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_613
tff(fact_10127_ATP_Olambda__614,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: fun_int_Code_integer,Uuc: int] :
      ( pp(aa_int_bool(aa_fun3610984093196804608t_bool(aa_fun2519502641837456723t_bool(aTP_Lamm_bs(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uuc)),aa_int_Code_integer(Uub,Uuc)) != one_one_Code_integer ) ) ) ).

% ATP.lambda_614
tff(fact_10128_ATP_Olambda__615,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_ca(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_615
tff(fact_10129_ATP_Olambda__616,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_co(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_complex_complex(plus_plus_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_616
tff(fact_10130_ATP_Olambda__617,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_cw(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_real_real(plus_plus_real(aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_617
tff(fact_10131_ATP_Olambda__618,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_ci(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_complex_complex(plus_plus_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_618
tff(fact_10132_ATP_Olambda__619,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_cq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_real_real(plus_plus_real(aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_619
tff(fact_10133_ATP_Olambda__620,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_cy(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_620
tff(fact_10134_ATP_Olambda__621,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_ck(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_complex_complex(plus_plus_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_621
tff(fact_10135_ATP_Olambda__622,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_cs(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_real_real(plus_plus_real(aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_622
tff(fact_10136_ATP_Olambda__623,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_da(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_623
tff(fact_10137_ATP_Olambda__624,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_cm(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_complex_complex(plus_plus_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_624
tff(fact_10138_ATP_Olambda__625,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_cu(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_real_real(plus_plus_real(aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_625
tff(fact_10139_ATP_Olambda__626,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ta(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_626
tff(fact_10140_ATP_Olambda__627,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_tg(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_627
tff(fact_10141_ATP_Olambda__628,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_tc(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_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_628
tff(fact_10142_ATP_Olambda__629,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_te(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_629
tff(fact_10143_ATP_Olambda__630,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_rd(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_630
tff(fact_10144_ATP_Olambda__631,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_rj(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_631
tff(fact_10145_ATP_Olambda__632,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_rf(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_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_632
tff(fact_10146_ATP_Olambda__633,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_rt(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_633
tff(fact_10147_ATP_Olambda__634,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_rh(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_634
tff(fact_10148_ATP_Olambda__635,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_ey(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uuc)) ).

% ATP.lambda_635
tff(fact_10149_ATP_Olambda__636,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_eu(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uuc)) ).

% ATP.lambda_636
tff(fact_10150_ATP_Olambda__637,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_ew(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uuc)) ).

% ATP.lambda_637
tff(fact_10151_ATP_Olambda__638,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_gv(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_638
tff(fact_10152_ATP_Olambda__639,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_gu(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_639
tff(fact_10153_ATP_Olambda__640,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_lt(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_640
tff(fact_10154_ATP_Olambda__641,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_xl(Uu),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_641
tff(fact_10155_ATP_Olambda__642,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_xr(Uu),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_642
tff(fact_10156_ATP_Olambda__643,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_xn(Uu),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uub),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_643
tff(fact_10157_ATP_Olambda__644,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_xp(Uu),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).

% ATP.lambda_644
tff(fact_10158_ATP_Olambda__645,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_sw(Uu),Uua),Uub),Uuc),Uud) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_sv(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_645
tff(fact_10159_ATP_Olambda__646,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_su(Uu),Uua),Uub),Uuc),Uud) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_st(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_646
tff(fact_10160_ATP_Olambda__647,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_ss(Uu),Uua),Uub),Uuc),Uud) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_sr(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_647
tff(fact_10161_ATP_Olambda__648,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_sq(Uu),Uua),Uub),Uuc),Uud) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_sp(Uua),Uub),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_648
tff(fact_10162_ATP_Olambda__649,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_yq(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_yp(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(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_649
tff(fact_10163_ATP_Olambda__650,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_th(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_tg(Uua),Uuc),Uud)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud)) ).

% ATP.lambda_650
tff(fact_10164_ATP_Olambda__651,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_tf(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_te(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_651
tff(fact_10165_ATP_Olambda__652,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_td(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_tc(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_652
tff(fact_10166_ATP_Olambda__653,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_tb(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ta(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_653
tff(fact_10167_ATP_Olambda__654,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_qd(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_654
tff(fact_10168_ATP_Olambda__655,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_qh(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Uud)),Uua)),one_one_real)),Uud)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_655
tff(fact_10169_ATP_Olambda__656,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_qf(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Uud)),Uua)),one_one_rat)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_656
tff(fact_10170_ATP_Olambda__657,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_qc(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_657
tff(fact_10171_ATP_Olambda__658,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_qg(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Uu)),Uua)),Uud)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_658
tff(fact_10172_ATP_Olambda__659,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_qe(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,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_659
tff(fact_10173_ATP_Olambda__660,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_up(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_660
tff(fact_10174_ATP_Olambda__661,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_ur(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(uminus_uminus_real,Uua)),Uud)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Uub)),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_661
tff(fact_10175_ATP_Olambda__662,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_uq(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(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(aa_rat_fun_rat_rat(plus_plus_rat,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_662
tff(fact_10176_ATP_Olambda__663,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: complex,Uuc: nat,Uud: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_sp(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_663
tff(fact_10177_ATP_Olambda__664,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_sv(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uuc)) ).

% ATP.lambda_664
tff(fact_10178_ATP_Olambda__665,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_sr(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(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_665
tff(fact_10179_ATP_Olambda__666,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_st(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_666
tff(fact_10180_ATP_Olambda__667,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_zk(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)))
        & ! [I: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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,I)),X_13))
            <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Uub),I)) ) )
        & ( ( Uuc = Uud )
         => ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Uua)))
             => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),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))
            & ! [X4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu)))
               => ( 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),X4))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),X4))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Uud)) ) ) ) ) ) ) ) ).

% ATP.lambda_667
tff(fact_10181_ATP_Olambda__668,axiom,
    ! [Uu: real,Uua: complex] : aa_complex_real(aTP_Lamm_eb(Uu),Uua) = Uu ).

% ATP.lambda_668
tff(fact_10182_ATP_Olambda__669,axiom,
    ! [Uu: real,Uua: int] : aa_int_real(aTP_Lamm_ec(Uu),Uua) = Uu ).

% ATP.lambda_669
tff(fact_10183_ATP_Olambda__670,axiom,
    ! [Uu: rat,Uua: complex] : aa_complex_rat(aTP_Lamm_dz(Uu),Uua) = Uu ).

% ATP.lambda_670
tff(fact_10184_ATP_Olambda__671,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_dy(Uu),Uua) = Uu ).

% ATP.lambda_671
tff(fact_10185_ATP_Olambda__672,axiom,
    ! [Uu: rat,Uua: int] : aa_int_rat(aTP_Lamm_ea(Uu),Uua) = Uu ).

% ATP.lambda_672
tff(fact_10186_ATP_Olambda__673,axiom,
    ! [Uu: nat,Uua: complex] : aa_complex_nat(aTP_Lamm_ef(Uu),Uua) = Uu ).

% ATP.lambda_673
tff(fact_10187_ATP_Olambda__674,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_eh,Uu),Uua) = Uu ).

% ATP.lambda_674
tff(fact_10188_ATP_Olambda__675,axiom,
    ! [Uu: nat,Uua: int] : aa_int_nat(aTP_Lamm_eg(Uu),Uua) = Uu ).

% ATP.lambda_675
tff(fact_10189_ATP_Olambda__676,axiom,
    ! [Uu: int,Uua: complex] : aa_complex_int(aTP_Lamm_ee(Uu),Uua) = Uu ).

% ATP.lambda_676
tff(fact_10190_ATP_Olambda__677,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_ed(Uu),Uua) = Uu ).

% ATP.lambda_677
tff(fact_10191_ATP_Olambda__678,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_an,Uu) = Uu ).

% ATP.lambda_678
tff(fact_10192_ATP_Olambda__679,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_ao,Uu) = Uu ).

% ATP.lambda_679
tff(fact_10193_ATP_Olambda__680,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ap,Uu) = Uu ).

% ATP.lambda_680
tff(fact_10194_ATP_Olambda__681,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_aq,Uu) = Uu ).

% ATP.lambda_681
tff(fact_10195_ATP_Olambda__682,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ar,Uu) = Uu ).

% ATP.lambda_682
tff(fact_10196_ATP_Olambda__683,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_as,Uu) = Uu ).

% ATP.lambda_683
tff(fact_10197_ATP_Olambda__684,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_ai,Uu) = zero_zero_complex ).

% ATP.lambda_684
tff(fact_10198_ATP_Olambda__685,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_aj,Uu) = zero_zero_real ).

% ATP.lambda_685
tff(fact_10199_ATP_Olambda__686,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_ak,Uu) = zero_zero_rat ).

% ATP.lambda_686
tff(fact_10200_ATP_Olambda__687,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_hw,Uu) = zero_zero_complex ).

% ATP.lambda_687
tff(fact_10201_ATP_Olambda__688,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_dd,Uu) = zero_zero_real ).

% ATP.lambda_688
tff(fact_10202_ATP_Olambda__689,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_al,Uu) = zero_zero_nat ).

% ATP.lambda_689
tff(fact_10203_ATP_Olambda__690,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_hx,Uu) = zero_zero_int ).

% ATP.lambda_690
tff(fact_10204_ATP_Olambda__691,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_am,Uu) = zero_zero_int ).

% ATP.lambda_691
tff(fact_10205_ATP_Olambda__692,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_wp,Uu))
    <=> $false ) ).

% ATP.lambda_692
tff(fact_10206_ATP_Olambda__693,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_wq,Uu))
    <=> $true ) ).

% ATP.lambda_693

% Helper facts (63)
tff(help_fNot_2_1_U,axiom,
    ! [P2: bool] :
      ( pp(P2)
      | pp(aa_bool_bool(fNot,P2)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P2: bool] :
      ( ~ pp(aa_bool_bool(fNot,P2))
      | ~ pp(P2) ) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fconj(P2),Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fconj(P2),Q))
      | pp(P2) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(P2)
      | ~ pp(Q)
      | pp(aa_bool_bool(fconj(P2),Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fdisj(P2),Q))
      | pp(P2)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P2: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(fdisj(P2),Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(P2)
      | pp(aa_bool_bool(fdisj(P2),Q)) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P2),Q))
      | ~ pp(P2)
      | pp(Q) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [Q: bool,P2: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P2),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(P2)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P2),Q)) ) ).

tff(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_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(aa_int_fun_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__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(if_rat(fTrue),X),Y) = X ).

tff(help_fEx_1_1_fEx_001t__Nat__Onat_U,axiom,
    ! [P2: fun_nat_bool,X: nat] :
      ( ~ pp(aa_nat_bool(P2,X))
      | pp(aa_fun_nat_bool_bool(fEx_nat,P2)) ) ).

tff(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] : aa_real_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(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_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] : 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] : 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(fequal_Extended_enat(X,Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ pp(fequal_Extended_enat(X,Y))
      | ( X = Y ) ) ).

tff(help_fAll_1_1_fAll_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P2: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( ~ pp(fAll_VEBT_VEBT(P2))
      | pp(aa_VEBT_VEBT_bool(P2,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_fequal_2_1_fequal_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( X != Y )
      | pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(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(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] : if_Sum_sum_nat_nat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] : if_Sum_sum_nat_nat(fTrue,X,Y) = X ).

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] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_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] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_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,
    ! [P2: fun_bool_bool,Q: fun_VEBT_VEBT_bool,R2: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_3745307990312268622T_VEBT(P2,Q),R2) = aa_bool_bool(P2,aa_VEBT_VEBT_bool(Q,R2)) ).

tff(help_COMBS_1_1_COMBS_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool_001t__HOL__Obool_U,axiom,
    ! [P2: fun_VE2066785035971159626l_bool,Q: fun_VEBT_VEBT_bool,R2: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBS_6222329278419335549l_bool(P2,Q),R2) = aa_bool_bool(aa_VEB8047906998784800507l_bool(P2,R2),aa_VEBT_VEBT_bool(Q,R2)) ).

tff(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P2: bool] :
      ( ( P2 = fTrue )
      | ( P2 = 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,
    ! [P2: fun_fu7168253649841883962l_bool,Q: fun_VE3763956218935105199t_bool,R2: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_2885587777813823575T_VEBT(P2,Q),R2) = aa_fun_nat_bool_bool(P2,aa_VEB4832260192069861672t_bool(Q,R2)) ).

tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P2: fun_bo1841576699660343469l_bool,Q: fun_VEBT_VEBT_bool,R2: vEBT_VEBT] : aa_VEB8047906998784800507l_bool(cOMBB_3438363824644082366T_VEBT(P2,Q),R2) = aa_boo8970175072276269406l_bool(P2,aa_VEBT_VEBT_bool(Q,R2)) ).

tff(help_COMBC_1_1_COMBC_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001t__HOL__Obool_U,axiom,
    ! [P2: fun_VE7346385682144900493T_bool,Q: set_VEBT_VEBT,R2: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBC_771164880915498576T_bool(P2,Q),R2) = aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(P2,R2),Q) ).

% Conjectures (1)
tff(conj_0,conjecture,
    info = some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)) ).

%------------------------------------------------------------------------------